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              <text>The essay aims to present an introduction to some of Goldie’s work on rings and modules, assuming only a second-year undergraduate knowledge of ring theory. Specifically, we will focus on Goldie’s theory of the quotient ring of prime Noetherian rings, exploring the concepts and results that underpin his celebrated theorem. </text>
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              <text>We consider a stochastic Ising model evolving according to Glauber dynamics on a complete graph $\G$ with two clusters $\G_1$,$\G_2$. We analyse the metastability of the model and its dependence on the interaction coefficient $J^*$ between vertices $v\in G_1$, $w \in \G_2$. We also investigate the change to dynamics when a positive magnetic field with strength $h$ is introduced. In our study of the metastability we focus on the expected hitting time $\mathbb{E}[\tau^m_s]$. In \cite{Manzo}, the authors show that in the absence of deep cycles, provided we have good control on the typical paths from a $\X_m$, it is possible to derive asymptotic results concerning this value. In addition to asymptotic behaviour, we analyse the dynamics in the $n\to \infty$ limit via large deviation theory. Thus we provide weak convergence results for the Gibbs measure for under the mean-field assumption and identify changes to the dynamics below and above critical temperatures. </text>
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              <text>Planetary gearboxes play a critical role in wind turbine drivetrains, enabling the efficient conversion of low-speed, high-torque rotor motion into high-speed rotation for electrical generation. However, frictional and material losses within these gear systems can significantly reduce overall turbine performance, particularly under low-speed operating conditions. This study experimentally investigated the mechanical efficiency of a scaled-down, 3D-printed planetary gearbox designed to replicate typical wind turbine gear ratios. A custom test rig was developed, incorporating a 12 V DC motor driving the input shaft at 5 - 25 rpm, with torque and speed measured using a current sensor, load cell and two infrared proximity sensors. Results validated the theoretical 1:8 step-up gear ratio and showed that efficiency increased from 69% at 5 rpm to a peak of approximately 85% around 15 - 16 rpm before declining at higher speeds. These findings demonstrate that small-scale polymer planetary stages can replicate real gearbox behaviour and offer valuable insight into efficiency optimisation and material selection for larger-scale wind energy applications.</text>
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