It is well known that the real geometric series ${\sum\limits_{n = 0}^\infty{a{k^n} = a + ak + a{k^2} + }}$ … converges to a definite sum if the common ratio, k, is such that |k| < 1, the sum being ${a}\over{1-k}$. For example, if a = 1 and ${k ={{1}\over{2}}}$ we obtain the series ${{1} + {{1}\over{2}} + ... + {{1}\over{2}^n} + ...}$, whose partial sums are ${{1}, {{3}\over{2}}, {{7}\over{4}}, ..., 2 -{{1}\over{2}^n}}$, …, and these are clearly approaching the value 2 as n becomes larger and larger. As n → ∞, ${2} - {{1}\over{2}^n}{\unicode{x2192}} \,\,2$, in agreement with the formula ${{a}\over{1-k}} {=} {{1}\over{1}-{1\over2}}=2$.

In order for accused persons with disabilities to be able to access justice on an equal basis with others, equality of outcomes is important. However, in the past century, the limited approach to legal aid which focuses on processes has continually been applied by criminal justice system actors in response to legal aid challenges faced by accused persons with disabilities in Kenya. The major dilemma facing this approach is its emphasis on steps to be taken as opposed to the end result or goal. This paper seeks to explain that a shift towards an outcomes approach to legal aid for accused persons with disabilities has the potential of supporting innovation in Kenya’s criminal justice system and help close access to justice gaps that may exist. To achieve this paradigm shift, the African Disability Protocol has been employed as it promotes an integral development of legal aid justice that requires states parties to put in place specific outcomes-based laws for promoting the rights of accused persons in access to justice.

We present some less known variations of the the Vecten configuration and give purely geometric proofs for them. It is unlikely that these variations (and even proofs?) are new, probably just well-hidden in the literature. If a reader happens to know references for the variations discussed (or other geometric proofs), please let the authors know. At [1] the reader can find a dynamic webpage on our topic.

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.