We investigate the limiting spectral distribution of a noncentral unified matrix model defined by $\boldsymbol{\Omega}(\mathbf{X}) = ({(\mathbf{X}\mathbf{P}_1+\mathbf{A})(\mathbf{X}\mathbf{P}_1+\mathbf{A})'}/{n_1}) ({\mathbf{X}\mathbf{P}_2\mathbf{X}'}/{n_2})^{-1}$, where $\mathbf{X}=(X_{ij})_{p\times n}$ is a random matrix with independent and identically distributed real entries having zero mean and finite second moment. $\mathbf{A}$ is a $p\times n$ nonrandom matrix. The matrices $\mathbf{P}_1$ and $\mathbf{P}_2$ are projection matrices satisfying $\mathrm{rank}(\mathbf{P}_1)=n_1$, $\mathrm{rank}(\mathbf{P}_2)=n_2$, and $\mathbf{P}_1\mathbf{P}_2=0$. When $\mathbf{P}_1$ and $\mathbf{P}_2$ are random, they are assumed to be independent of $\mathbf{X}$. When $p/n_1\to c_1\in(0,\infty)$ and $p/n_2\to c_2\in(0,1)$, we establish the almost sure convergence of the empirical spectral distribution of $\boldsymbol{\Omega}$ to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.

What does the institutionalisation of a protest movement into an opposition party in an electoral autocracy mean for its members? This article examines this question by analysing the conversion of People Power, a political pressure group in Uganda, into the National Unity Platform (NUP), focusing on the dilemmas of organisation, strategy and identity. NUP sought to broaden and institutionalise People Power’s activities to be seen as a credible party capable of holding state power. Simultaneously, its initial political weight was closely tied to the defiant, extra-parliamentary energy of the grassroots – a resource the party needed to preserve. These tensions were intensified by Uganda’s authoritarian context, where state repression and demobilisation intersected with uneven access to resources and patronage, producing frictions between privileged actors and grassroots members. The paper shows how these dilemmas generated frustrations among bottom-up constituencies and highlights the importance of examining intra-party processes from a grassroots perspective.

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X. We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H-orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G, and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H-orbit to a single compact set in X. This complements our results in the case of an arithmetic lattice.

Modern translators and commentators have uniformly taken the phrase καὶ τιμωμένων ἀντετιμᾶτο in Max. Tyr. Or. 3.2 as a reference to Socrates’ reported proposal of a counter-penalty as depicted in the second speech of Plato’s Apology. This article suggests an alternative interpretation rooted in both the surrounding context of Or. 3 and an analysis of Greek forensic vocabulary and usage. The latter analysis also serves to cast doubt on the claim, common in discussions of Athenian law, that ἀντιτιμᾶσθαι served as the technical term for making a counter-penalty proposal.

The depth of a subgroup H of a finite group G is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb {C}H \subseteq \mathbb {C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

Pleasure is widely thought to have intrinsic value. However, this thesis has been threatened by the argument that pleasure is a mental state that essentially involves the subject’s conative attitudes. Its value, then, would be subjective. Though the existing version of the argument can be resisted by simply rejecting the attitudinal theories of pleasure on which it is based, I will develop a new and more general version based on the reasonable hypothesis that the phenomenal character of pleasure is reducible to a physical or functional property. If this new version is convincing, then the most promising way to secure the intrinsicality of the value of pleasure and to escape all versions of the subjectivity argument might be to embrace a non-reductionist account of pleasure and its value.

Let $S_k$ denote the space of cusp forms of weight k and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi $ mod D, we introduce twisted periods $r_{t,\chi }$ on $S_k$. We show that for a fixed natural number n, if k is sufficiently large relative to n and D, then any n periods with the same twist but different indices are linearly independent. We also prove that if k is sufficiently large relative to $D,$ then any n periods with the same index but different twists mod D are linearly independent. These results are achieved by studying the trace of the products and Rankin–Cohen brackets of Eisenstein series of level D with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda’s conjecture implies a non-vanishing result on twisted central L-values of normalized Hecke eigenforms.

Let f be a rational map of degree $d\geq 2$. The moduli space $\mathcal {M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting of all quasiconformal conjugacy classes of f. For f that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal {M}_f\to X_f$ such that $i(\mathcal {M}_f)$ is precompact in $X_f$. In particular, $\mathcal {M}_f$ is Carathéodory hyperbolic (that is, bounded holomorphic functions separate points in $\mathcal {M}_f$), provided that f is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.

Marine protected areas (MPAs) have proliferated to protect marine ecosystems and manage unsustainable fishing, but their outcomes vary by economic and governance contexts. Drawing on a panel dataset spanning 1995–2021 and employing the Pressure–State–Response framework, this study analyses how MPA coverage is associated with overexploited fish stocks and examines Official Development Assistance (ODA). Results show MPAs in high-income countries are associated with lower overexploited catch rates, reflecting robust governance. In low-income nations, however, limited capacity is often linked to ‘paper parks’ with negligible impact. Multi-purpose ODA – supporting sustainability goals – appears more effective than single-purpose ODA and is associated with better conservation outcomes. Integrating MPAs with fisheries management, supported by international assistance for enforcement, appears important for bridging disparities in effectiveness. By highlighting the interplay among economic conditions, governance and funding, this study offers higher-level insights into factors that shape MPA effectiveness, contributing to broader policy discussions on marine conservation.

Clozapine remains the gold-standard medication for treatment-resistant schizophrenia, yet its use in young people is limited despite emerging evidence supporting efficacy. In this journal, Walker et al highlight the unmet need for training and education on clozapine prescribing in child and adolescent populations, reinforcing concerns raised by leading experts about real-world implementation barriers. In this commentary on their article, we consider some of these concerns, particularly regarding therapeutic benefit and metabolic and haematological safety. We conclude that, given the severe consequences of treatment delay and the prognostic burden associated with early-onset schizophrenia, a proactive, algorithmic and well-supervised approach to clozapine initiation in youth is essential.

Since the 1920s, the workers’ movement has been crucial in linking international communism with colonial liberation. From the 1950s, communist trade unions and the World Federation of Trade Unions (WFTU) played a key role in this process. The WFTU forged ties with Afro-Asian unions to challenge Western powers with an anti-imperialist platform. Decolonization opened new opportunities for European workers in their anti-capitalist struggle, prompting unions such as the French Confédération Générale du Travail (CGT) and Italian Confederazione Generale Italiana del Lavoro (CGIL) to build relationships with unions in newly independent African nations. Their focus was mainly on North and West Africa, where socialist movements emerged during the late 1950s and early 1960s. The CGT and the CGIL established solid links with trade unions in Guinea, Mali, and Ghana. Their views, shaped by their affiliations with the French and Italian communist parties, were not consistently aligned. The CGT maintained a Eurocentric approach to African socialism, while the CGIL, aligned with the PCI, supported a “staged” revolution in Africa, where the working class did not occupy a position of leadership. The CGIL and PCI emphasized the role of peasants and workers in building socialism through a “social revolution” that could strengthen the socialist bloc. These differences created tensions in African trade unions and the FSM, leading to the marginalization of the CGIL’s approach. Nonetheless, Italian unionists gained support from African unions and joined the CGT in mediating for workers in Guinea, Mali, and Ghana.

Obsessive-compulsive disorder (OCD) is a complex psychiatric disorder. While existing studies have revealed abnormalities in brain structure and function associated with OCD, there is a paucity of research integrating these two aspects, and the transcriptional patterns underlying these abnormalities remain unclear. This study is a multiscale, exploratory investigation designed to generate hypotheses rather than to test causal mechanisms. We aimed to investigate aberrations in brain structure–function coupling (SFC) in OCD patients and, by integrating gene expression profiles and neurotransmitter maps, to explore the potential molecular and genetic bases of these changes. We recruited 100 medication-free OCD patients and 90 healthy controls, and employed multimodal imaging techniques to systematically analyze abnormalities in static SFC in OCD patients. Subsequently, we conducted transcriptomic analysis to identify genes associated with SFC abnormalities and performed spatial correlation analysis with neurotransmitter atlases to investigate potential links between SFC dysregulation and transcriptional patterns. Our findings demonstrated that OCD patients exhibit significant SFC abnormalities in the right temporoparietal junction (rTPJ). These SFC abnormalities are significantly associated with 2,421 gene expression profiles and the serotonin neurotransmitter system. Gene enrichment analysis revealed that these aberrant genes are primarily involved in key biological processes, such as brain development, synaptic signaling, cell projection development, and regulation of neuronal processes. By integrating multimodal imaging, transcriptomic, and neurotransmitter data, this study provides multiscale evidence for the potential molecular basis of SFC abnormalities in the rTPJ of OCD patients, offering preliminary insights into a possible pathological pathway of OCD.

Many believe that relationships can make a constitutive difference to the moral status of paternalistic treatment. For example, it is often assumed that it’s easier to justify paternalizing a spouse than a stranger. But although this thought is widespread, there exists no detailed account of how relationships could mitigate paternalistic complaints. The aim of this paper is to develop an account of this phenomenon, drawing on the work of Margaret Gilbert and the notion of joint commitments. According to the resulting view, close relations can constitutively mitigate paternalistic complaints by rendering paternalistic interference consistent with the will of the paternalized agent.

From classical trigonometric formulas, complex differential equations of various types have been formulated and widely studied. We investigate perturbed nonlinear complex differential equations and explore their corresponding complex differential systems. Some open questions on nonlinear complex differential equations and systems are proposed.