Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs.

A $(p,q)$-colouring of a graph $G$ is an edge-colouring of $G$ which assigns at least $q$ colours to each $p$-clique. The problem of determining the minimum number of colours, $f(n,p,q)$, needed to give a $(p,q)$-colouring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in \{4,5\}$, each of which was proved by giving a deterministic construction which combined a $(p,p-1)$-colouring using few colours with an algebraic colouring.

In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colourings of $p$-cliques with $p-1$ colours which can appear in our modified version of the $(p,p-1)$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$-colourings, which would otherwise make this problem intractable for large values of $p$. In addition, we generalize our algebraic colouring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$.

Hyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks. In particular, they have a power-law degree distribution with controllable exponent $\beta$ and high clustering that can be controlled via the temperature $T$.

We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to $T = 0$. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.

Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.

Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases.

The proof theory of the constructive modal logic S4 (hereafter $\mathsf{CS4}$) has been settled since the beginning of this century by means of either standard natural deduction and sequent calculi or by the reconstruction of modal logic through hypothetical and categorical judgments à la Martin-Löf, an approach carried out by using a special kind of sequents, which keeps two separated contexts representing ordinary and enhanced hypotheses, intuitively interpreted as true and valid assumptions. These so-called dual-context sequents, originated in linear logic, are used to define a natural deduction system handling judgments of validity, truth, and possibility, resulting in a formalism equivalent to an axiomatic system for $\mathsf{CS4}$. However, this proof-theoretical study of $\mathsf{CS4}$ lacks, to the best of our knowledge, its third fundamental constituent, namely a sequent calculus. In this paper, we define such a dual-context formalism, called ${\bf DG_{CS4}}$, and provide detailed proofs of the admissibility for the ordinary cut rule as well as the elimination of a second cut rule, which manipulates enhanced hypotheses. Furthermore, we make available a formal verification of the equivalence of this proposal with the previously defined axiomatic and dual-context natural deduction systems for $\mathsf{CS4}$, using the Coq proof-assistant.

Digital identity systems are promoted with the promise of great benefit and inclusion. The case of the Ugandan digital identity system demonstrates that the impact of digital identity systems is not only positive but also has negative impacts, significantly affecting human lives for the worse. The impact on the human lives of digital identity systems can be assessed by multiple frameworks. A specific framework that has been mentioned is the capabilities approach (CA). This article demonstrates that the CA is a framework to assess the impact on human lives that can be operationalized for technology and information and communication technology, including digital identity systems. Further research is required to compare the CA with other candidate evaluation frameworks.

This paper provides an examination of inter-organizational collaboration in the UK research system. Data are collected on organizational collaboration on projects funded by four key UK research councils: Arts and Humanities Research Council, Economic and Social Research Council, Engineering and Physical Sciences Research Council, and Biotechnology and Biological Sciences Research Council. The organizational partnerships include both academic and nonacademic institutions. A collaboration network is created for each research council, and an exponential random graph model is applied to inform on the mechanisms underpinning collaborative tie formation on research council-funded projects. We find that in the sciences, collaborative patterns are much more hierarchical and concentrated in a small handful of actors compared to the social sciences and humanities projects. Institutions that are members of the elite Russell Group (a set of 24 high-ranking UK universities) are much more likely to be involved in collaborations across research councils.