In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way providing income under specified bequest motive, from defined contribution pension pot. A key feature present is that it relaxes assumption fixed proportions invested in tontine accounts. making proportion an additional control function obtain, hitherto unavailable, closed-form solut...

Volatile binary liquid samples on wetting substrates are known to undergo either spreading or contraction tendencies, as a result of solutal Marangoni stresses due differential volatility. Enhanced is commonly thought occur when the lower surface tension component more ‘volatile’, while expected otherwise. We seek test limits this scenario for various configurations such sessile drops with free...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

The formulation of the classical basic equations for a thermoelectric cooler from the Thomson relations to the non-linear differential equation with Onsager’s reciprocal relations was performed to basically study the Thomson effect in conjunction with the ideal equation. The ideal equation is obtained when the Thomson coefficient is assumed to be zero. The exact solutions derived for a commerci...