Introduction to LSTM networks
This post is intended as a brief introduction to recurrent neural networks, focusing on long-short term memory cells, for those with a mathematical backgroun...
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Elementary Fractional Calculus
In this note we define the fractional derivative \(\frac{\mathrm{d}^\alpha f}{\mathrm{d}x^\alpha}\) of a function \(f\), where \(\alpha\) is allowed to be a...
The Bessel Equation
In this note we investigate series solutions to the Bessel equation \begin{equation} \label{eq:bessel} z^2 u'' + zu' + (z^2 - \nu^2)u = 0 \end{equation} So...
Automorphisms of the Upper Half Plane
In this note we study the group \(\mathrm{Aut}(\mathbb{H})\) of analytic automorpisms of the upper half plane. We review the usual classification of such au...
Examples of Neural Adaptation
In this note we investigate how the firing rate response of certain neurons adapts to a constant input stimulus. In particular, we look at the neural adaptiv...