The placebo effect is a genuine psychobiological phenomenon in which the expectation of improvement can lead to actual changes, including alterations in perception, behaviour and physiological responses. This article explores this phenomenon by dispelling common misconceptions and highlighting its significance for critical thinking.

Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$. We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times }$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm {Br}(L)[p]$ but the mod p Massey product $\langle a,b,c,d\rangle$ is not defined over $L$. Thus, the strong Massey vanishing conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^{* }(\Gamma _L,\mathbb Z/p\mathbb Z)$ of the absolute Galois group $\Gamma _L$ of $L$ is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.

We analyze the existence of Kähler–Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity (X,p). This boils down to solving a Dirichlet problem for certain complex Monge–Ampère equations. We establish a Moser–Trudinger inequality $(MT)_{\gamma}$ in subcritical regimes $\gamma<\gamma_{\rm crit}(X,p)$ and show the existence of smooth solutions in those cases. We show that the expected critical exponent $\tilde{\gamma}_{\rm crit}(X,p)=(({n+1})/{n}) \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.

In his Epistola de harmonica institutione (c.900 CE), Regino of Prüm names fourteen antiphons that he calls nothae – that is, ‘degenerate and illegitimate – that begin in one mode, are yet another in the middle, and end in a third’. These antiphons represent two different types of modulation: one diatonic, the other resulting from systemic transposition brought about by chromatic alteration. A rationale for both types of modulation is offered by the Musica and Scolica enchiriadis, respectively, both dating to the second half of the ninth century, with the Scolica providing a theory of vitia, or ‘corruptions’, to accommodate chants modulating by means of chromatic alteration. Modulation likewise played an important role in Eastern chant. Gerda Wolfram has shown that both diatonic and chromatic modulation can be documented in the earliest manuscripts of Byzantine chant, namely those dating to the tenth century. Indeed, the Hagiopolites, the oldest preserved Byzantine treatise on music (twelfth century CE), discusses chromatic modulation via what are called phthoraí (‘corruptions’), like the vitia in the West, and the papadikaí, or singers’ manuals, explicate the theory of diatonic modulation called ‘parallagḗ’. This article illustrates both phthorá and parallagḗ with an exercise from the treatise on church music by Akakios Chalkeopulos (c.1500 CE), and concludes that not just the nomenclature and intonation formulas of the Byzantine modes, but also the technique of modulating within a single chant were features shared by both Eastern and Western chant already in the earliest stages of their respective written traditions.

An Arabic-language tract crafted in in Makhachkala in 1949 offered an abrasive critique of ‘Alī al-Ghumuqī (1878–1943), ostensibly the father of the Dagestani modernist milieu (al-firqa al-jadidiyya). Who was ‘Alī al-Ghumuqī, what was his oeuvre, and why did the most prominent ulama of Dagestan despise him to the extent of publishing an original pamphlet cursing his legacy? In this article we set out to answer these questions and attempt to show that at the beginning of the Soviet century, the North Caucasus represented an important conduit for the circulation and further refinement of Islamic scholarship. We contend that the absorption and reproduction of modernist thinking among Dagestani ulama was not halted by the October Revolution and the Bolsheviks’ takeover. Indeed, we set out to show that in the North Caucasus between the 1920s and the 1960s, scholars continued to cultivate interest in Islamic jurisprudence, in fact unencumbered by the secularist policies adopted by the Soviet state. As we shall see, in this environment ‘Alī al-Ghumuqī morphed into what could be termed an epic figure and became so popular as to personify either the virtues or the evil aspects of modernist Islam.

Auditory-based illusions and effects are fascinating fields for both psychoacoustic research and sound installations. While such illusions and effects are usually researched in isolated scientific studies, they can also be applied as compositional tools in sound installations. This article addresses the suspenseful connection between psychoacoustic research and sound installations. After defining terms relevant to auditory-based illusions and effects, various aspects of sound installations are described. In that light, auditory-based illusions and effects are described and categorised and examples are provided for their scientific investigation by means of references to key experiments. Further, examples of applications are included that showcase the use of auditory-based illusions and effects in compositions and sound installations. Finally, in order to foster future artistic applications, the connections between illusions and effects are visualised, and sound-installation aspects are provided in a table. Such a combined consideration of psychoacoustic fundamentals and sound-installation aspects aims not only at deepening the methodological knowledge of sound artists, but also inspiring innovative compositional perspectives.

What is climate history? How can it serve as a lens through which to view other historical questions? This roundtable identifies key themes in Gilded Age and Progressive Era climate history, and demonstrates that this era was pivotal for both scientific and cultural perceptions of climate. It also shows how climate history can illuminate other subjects, including histories of science, medicine, health, and race. Further, it considers present-day implications. This roundtable began as a session sponsored by the Society for Historians of the Gilded Age and Progressive Era at the 2024 Organization of American Historians annual meeting in New Orleans. What follows is a conversation based on that panel, a selected bibliography of scholarly sources, and a collection of primary sources for teaching climate history.

While States can often refer to a single language text of a multilingual treaty, there are times when an examination of other language texts is required. This article proposes a novel three-step method for applying Article 33(4) of the Vienna Convention on the Law of Treaties to remove, or otherwise reconcile, differences in meaning between multilingual treaty texts. In doing so, this article seeks to address the current vacuum of practical guidance on when an examination of different authentic treaty texts is necessary in the process of interpretation, and how any differences in meaning between the texts should be removed or reconciled.

One set of policies that have been suggested to address climate change are carbon prices. Pricing carbon makes it more expensive to make polluting or unsustainable choices and less expensive to make cleaner or sustainable choices. This article explains why, if well designed, carbon prices can provide systematic signals to the market and make societies fairer. The article also defends carbon prices from several objections.

This article explores how capital-labor relations were conceptualized in late nineteenth-century protectionist thought. Taking as an example the American Protective Tariff League (APTL), a national protectionist pressure group that was heavily influenced by industrial interests and attempted to popularize protectionist ideas by issuing newspapers, pamphlets, leaflets, and posters, it reconstructs the arguments protectionist industrialists used in their agitation targeted at industrial workers. Following the protectionist wage argument, the APTL made the supposed wage benefit to laborers in protected industries the center of their argument. This wage argument was strongly intertwined with nativist and Anglophobic stereotypes. Further, the APTL proposed a unity of interests between capital and labor in tariff matters that hinged on a nationalist interpretation of economic matters, in which the American national economy was conceptualized as being endangered by imports and competition from other national economies but simultaneously as a harmonious cooperation of capital and labor on the inside. Analyzing the organized labor movement’s response to such claims, the article argues that this sort of agitation, while important to industrialists’ arguments, probably had little influence on workers and their stance on the tariff issue.