It is well known that almost all graphs are canonizable by a simple combinatorial routine known as colour refinement, also referred to as the 1-dimensional Weisfeiler–Leman algorithm. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of colour refinement and vertex individualization yields a canonical labelling for almost all circulant digraphs (i.e., Cayley digraphs of a cyclic group). This result provides first evidence of good average-case performance of combinatorial refinement within the class of vertex-transitive graphs. Remarkably, we do not even need the full power of the colour refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are compact in the sense of Tinhofer, that is, their polytops of fractional automorphisms are integral. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the more powerful 2-dimensional Weisfeiler–Leman algorithm.

This paper is concerned with Liouville-type theorems of positive weak solutions to elliptic $m$-Laplace equation and inequality with the logarithmic nonlinearity $u^q(\log u)^p (q,p\geqslant0)$. Using a direct Bernstein method we obtain a first range of values of $m,q,p$ in which all positive weak solutions of equation are constants, this holds in the following cases: (i) $1 \lt m \lt 2$, $m-1 \lt q \lt 1$, $0 \lt p \lt q$; (ii) $m \gt 1$, $q\geqslant1$, $0 \lt p \lt 1$. When $q=1$, the positive weak solutions are required to be bounded. Based on transformation of inequality and the utilization of suitable cut-off functions, we establish a Liouville-type theorem for positive weak solutions of inequality; this result also remains valid on complete noncompact Riemannian manifold.

Debates over the links between ethnicity and conflict often focus on the national level and take an ahistorical approach. This approach hides cases of ethnic conflict that arise at the subnational level and leaves unanswered questions over how ethnicity became a driver of conflict. This article explores these blind spots, using three cases in the African Great Lakes region. The cases reviewed here are the bipolar situations of Hema v. Lendu in Ituri (DRC), Banyarwanda/Banyamulenge v. ‘Autochthons’ in South and North Kivu (DRC), and Hima v. Iru in Ankole (Uganda). These cases suggest that polarisation is a more useful approach than fragmentation, but simple correlations between ethnic dyads and conflict obfuscate the nature and depths of the cleavages, as well as the mechanisms fuelling them. We elaborate on the pathways of escalation, highlighting how and when elite manipulations can activate deeply held identitarian norms. We conclude by emphasising the many lulls and moments of de-escalation, countering the portrayal of ethnic conflict as somehow inevitable.

After nearly two decades of documenting Ngũgĩ wa Thiong’o’s life and work, this reflection situates my conversations with him within a broader canvas—one that allows us to hear him in dialogue with fellow writers, activists, and global artists.

To what extent does refugee protection in Western Europe depend on the ethno-religious and gender identity of asylum seekers? This article examines how selective humanitarianism, shaped by the identity of asylum seekers and migrants, shapes their protection status. It offers an analysis of Germany’s response to Yezidi refugees, in comparison with that of France, in the wake of the genocidal campaign carried out by the Islamic State in 2014. Drawing on fieldwork that includes interviews with Yezidi refugees and stakeholders in Germany, we argue that contemporary asylum regimes operate through three interrelated mechanisms: the securitization of certain groups, selective humanitarian exceptions, and neoliberal selection criteria. The Yezidi experience illustrates how these mechanisms generate hierarchies of protection, wherein even recognized victims of genocide must meet increasingly economic thresholds to secure lasting refuge. While specialized programs for women survivors represent important humanitarian innovations, they often exclude male family members, thereby producing new forms of vulnerability. Struggling to align with dominant narratives of economically valuable migrants, Yezidis encounter a renewed form of liminality in Europe.

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$, we recover the conditions given in [7] for the existence of a unital partial action of $G$ on $R$.

Le nouveau réalisme développé par Maurizio Ferraris fait de Immanuel Kant son adversaire privilégié. Celui-ci aurait mis à distance le réel au travers de schèmes conceptuels et ouvert la postmodernité qui ne pense qu’à l’aune du corrélationisme et du constructivisme. Pourtant, Kant est essentiel à sa pensée et plus que d’une opposition, il s’agit pour Ferraris de renverser l’oeuvre kantienne en trouvant dans la Critique de la faculté de juger une ontologie naturelle émergentiste, et dans la Critique de la raison pure une ontologie sociale reposant sur la documentalité.