For integers n, , let Sk (n) denote the power sum 1k + 2k +… + nk, with S0 (n) = n. As is well known, Sk (n) can be represented by a polynomial in n of degree k + 1 with constant term equal to zero, i.e. ${S_k}\,(n)\, = \,\mathop \sum \nolimits_{j\, = \,1}^{k\, + \,1} \,{a_{k,j}}{n^j}$, for certain rational coefficients ak,j. In 1973, in the popular science magazine Kvant*, appeared an article by Vladimir Abramovich [1] in which, among other things, the author derived the following minimal recurrence relation which expresses in a very compact way the connection between the power sums Sk (n) and Sk-1 (n):

(1)

where the polynomial $S_{k\, - \,1}^*\,(n)$ is constructed by replacing in Sk-1 (n) the term nj with the expression ${{{n^{j + 1}} - n} \over {j + 1}}$ for each j = 1, 2, … , k. Note that $S_{k\, - \,1}^*\,(n)$ has degree k + 1, as it should be.

At university I was intrigued by a historian friend and his constant quest for gobbets – short, apposite quotations that he avidly collected for possible use in future essays. It strikes me that there might be analogous nuggets of mathematics and I offer the following five short, self-contained items both for amusement and for possible classroom use. Each contains a surprising element and I have deliberately left a few loose ends to encourage further exploration.

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.