Let ABC be a triangle with sides a, b, c, semiperimeter s, circumradius R, inradius r and area Δ. We introduce

$$Q\, = \,{(a - b)^2}\, + \,{(b - c)^2}\, + \,{(c - a)^2}$$

$$M\, = \,{(\left| {a - b} \right|\, + \left| {b - c} \right|\, + \,\left| {c - a} \right|)^2}.$$

Given an elliptic curve $ E $ over $ \mathbb {Q} $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ \chi $ of order $ q $, and in many cases its associated special L-value $ \mathscr {L}(E, \chi ) $ is known to be integral after normalizing by certain periods. This article determines the precise value of $ \mathscr {L}(E, \chi ) $ in terms of Birch–Swinnerton-Dyer invariants when $ q = 3 $, classifies their asymptotic densities modulo $ 3 $ by fixing $ E $ and varying $ \chi $, and presents a lower bound on the $ 3 $-adic valuation of $ \mathscr {L}(E, 1) $, all of which arise from a congruence of modular symbols. These results also explain some phenomena observed by Dokchitser–Evans–Wiersema and by Kisilevsky–Nam.

A multi-finger radio frequency (RF) transistor has been divided into multiple gate sections which can be biased independently. This provides a system designer the ability to dynamically reconfigure the output power and power gain of the device while maintaining good power efficiency and without changing the input drive power. By selectively switching the gate biases below pinch-off to effectively reduce the device’s active periphery, the maximum current of the device can be tuned to “follow” a reduced drain bias voltage, so that the optimum impedance at lower power remains similar to the one at full power, and a fixed matching network can be used to accommodate all power modes. The concept has been tested in a large signal load–pull characterization campaign on a test cell and implemented in a K-band power amplifier (PA) prototype. Measurements on the PA confirm the effectiveness of the method, achieving 30% efficiency at around 4.8–4.9 dB of output power tunability when maintaining a constant input power.

This article examines the notions of productivity and creativity with respect to complex verbs in English. Verb-forming suffixation involves the attachment of the suffixes ‑ize, ‑ify, -en and -ate to a base to form complex verbs such as hospitalize, densify, sharpen and hyphenate. Sampson (2016) describes productive processes that conform to existing patterns as F-creativity, or Fixed-creativity, and those that deviate from those patterns as E-creativity, or Enlarging/Extending creativity; Bergs (2018) and Uhrig (2018) view the F–E dichotomy as a cline. Coercion effects can account for linguistic productivity and creativity; Audring & Booij (2016) propose that the coercive mechanisms of Selection, Enrichment and Override lie on a unified continuum. This article integrates the F–E creativity and coercion continua, and analyses a database of conventionalized and recently coined complex verbs (Laws 2023) for instances of coercion. The results reveal that coercive mechanisms, particularly Selection and Enrichment, facilitate productivity and creativity in more complex constructional schemas underlying verbal derivatives, and that these coercive patterns have become increasingly more entrenched over time. E-creativity of complex verbs is defined here as ‘Unruly’ coercion and the nature of attested examples is discussed.

We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime p. The conjectures state that: (i) $J_p$ is always finite and of the order $O(p^2(\log \log p)^{2+\epsilon })$; (ii) the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; (iii) no harmonic number is divisible by $p^4$. We prove parts (i) and (iii) for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to $50\times 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately $30$ and $50$, respectively.

The relationship involving the unknown other has so far been exclusively translated into the language of fear as part of the securitised response to migration. The fear of the unknown other divides people into those who are associated with illegality and chaos and those who need to be protected from such ‘danger’. In contrast, the humanitarian approach to migration challenges the securitised response to the unknown other: it refuses to separate the self from the other and instead appeals to the idea of common humanity. This paper draws on the idea of the gothic to develop a humanitarian way of embracing the fear of the unknown. In the gothic framework, the other is feared not because of categorical differences between the self and the other, embodied in the securitised response to migration, but categorical ambiguity between the two. Using UK-based welcome activism as an example, I argue that gothic-inspired humanitarianism embraces the fear of the unknown other through the sharing of not knowing oneself. This offers a new basis for solidarity, in the language of fear, without resorting to the securitised relationship between the self and the other.

This article explores India’s ‘long wars’ – the counter-insurgency campaigns the state imposed on recalcitrant populations and territories. Existing critical debates have focused on colonial and imperial counter-insurgency waged by developed Western states and empires. Yet these powers hardly command a monopoly on how these are fought, rationalised, or imagined. Indian counter-insurgency campaigns are a key case in point. The aftermath of British colonial rule led to a revivification of rather than an end to counter-insurgency. Indian counter-insurgency thinking betrays similar logics of differentiation to those of the British. However, an engagement with Indian counter-insurgency archives reveals that the political economy of (post-)colonial rule results in its own particular sets of inclusions and exclusions. We tease out these tensions and anxieties that underpin India counter-insurgency by exploring how India’s long wars in its north-eastern states have been rationalised and explained away among Indian counter-insurgents, namely through references to ‘diversity’ and ‘democracy’. Such references index a politics premised on a disavowal of violence, which represents a weapon of war. This disavowal, narrated through exceptionalist claims, manifests itself through distinct modalities with their own tensions and even contradictions, leading to India’s own complicated relationships with notions and practices of coloniality.

We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree.

The centre of mass of a plane, uniform, lamina in the shape of a regular n-gon coincides with the centre of mass of n unit masses, one at each of its vertices. As many n-gons fail to have this property it is remarkable that all triangular laminas, whether regular (i.e. equilateral) or not, do have this property. Here we discuss this difference and we assume throughout that each lamina has unit density (so, disregarding units, its mass is the same as its area). We shall refer to the centre of mass of the lamina as its centroid, and the centre of mass of unit point masses, one at each vertex, as its vertex-centroid. Also, we shall say that a polgonal lamina is vertex-equivalent if its centroid coincides with its vertex-centroid, i.e. if, and only if, .

Does the political knowledge gender gap extend to knowledge about federalism, an institutional arrangement that increases the cognitive demand on voter knowledge? We answer this question by drawing upon data from three national surveys administered in Canada between 2020 and 2022. We find evidence of a gap between men and women in terms of their knowledge of the distribution of authority across the three orders of government. Across four of our knowledge items, the gender gap favouring men gets smaller as the issues vary from the federal to provincial to municipal level. Knowledge about national defence and sewage/water, however, do not fit this pattern. These results suggest future research should examine whether the gendered knowledge gap with respect to federalism can be explained by which levels of government have responsibility over areas of jurisdiction that have a strong effect on or are used by women on a daily basis.

Approaches to creativity commonly distinguish between F-creativity (rule-compliant use) and E-creativity (rule-breaking use). This dichotomy in part stems from a focus on grammatical constructions (‘nodes’) at the relative expense of their connections (‘links’). We approach creativity and productivity from a link-based perspective in Usage-Based Construction Grammar, and assume that productivity pertains to a unit’s inventory of links, while creativity pertains to the creation and maintenance of links. These assumptions are showcased using the into-causative (He talked me into going, They scared us into working harder). The construction is productive because it hosts a large inventory of verbal slot-fillers (talk, scare). Conversely, these slot-fillers themselves are creative because they can establish and maintain links with a construction that is not their primary host. This property is not linear: we assume that the slot-fillers’ ability to occur in unusual constructional environments reflects their general ‘creative potential’ to form and maintain (new) links within the network. In data from the Corpus of Contemporary American English (COCA), we find weak, but consistent correlations between verbs’ association with the into-causative and (i) their semantic and syntactic compatibility with the construction, and, crucially, (ii) their general flexibility and ability to establish and maintain links.