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              <text>We consider the infinite-horizon Merton investment-consumption problem. We begin by formally introducing the problem in the simplest, one-dimensional setting and show how one might derive a candidate solution. We then detail two different rigorous verification arguments showing this strategy is indeed the optimal one. After this, we move to a slightly more heuristic setting where we will consider two different variations of the problem - namely the case of transaction costs and non-constant parameters - and discuss to what extents the previous arguments extend to these more general situations. The report is written to be a relatively gentle introduction to the field of optimal stochastic control and continuous time finance, and to be accessible to any student with some reasonable prior understanding of stochastic calculus.</text>
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              <text>The Multi-Adaptive Threshold Model is a system of equations that describe an action potential for a neuron cell. What is special about this model is that the spiking threshold, which is the threshold of potential required for a neuron cell to spike, adapts with time instead of being constant. The external input current of the neuron is also a stochastic process, and therefore the neuron potential also isn’t deterministic. This project looks at the theoretical properties of the system of equations, plots the system of equations  features in the spiking times of neuron cells.  </text>
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              <text>The brown dwarf metallicity correlation was investigated for a set of 48 transiting brown dwarfs, taken from wide field surveys. Host star metallicity was compared to field stars using the Besançon galaxy model, with a metallicity offset between the sets found. Quantifying this using a power law the correlation constant was found to be B= 1.08+0.58 −0.52. Limiting the sample to those brown dwarfs below 42.5 MJ and above 42.5 MJ the correlation constants were found to be B= 1.44+0.87 −0.79 and B= 0.95+0.66 −0.54 respectively. This suggests the population of brown dwarfs with masses below 42.5 MJ form via core accretion whereas those with masses above 42.5 MJ form via gravitational collapse.</text>
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