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Simulating the Belousov-Zhabotinsky reaction

The Belousov-Zhabotinsky (BZ) reaction is a classical example of a non-equibrium chemical oscillator in which the components exhibit periodic changes in concentration.

The double pendulum

In classical mechanics, a double pendulum is a pendulum attached to the end of another pendulum. Its equations of motion are often written using the Lagrangian formulation of mechanics and solved numerically, which is the approach taken here. The dynamics of the double pendulum are chaotic and complex, as illustrated below.

Visualizing the bivariate Gaussian distribution

The multivariate Gaussian distribution of an $n$-dimensional vector $\boldsymbol{x}=(x_1, x_2, \cdots, x_n)$ may be written

Visualizing the gradient descent method

In the gradient descent method of optimization, a hypothesis function, $h_\boldsymbol{\theta}(x)$, is fitted to a data set, $(x^{(i)}, y^{(i)})$ ($i=1,2,\cdots,m$) by minimizing an associated cost function, $J(\boldsymbol{\theta})$ in terms of the parameters $\boldsymbol\theta = \theta_0, \theta_1, \cdots$. The cost function describes how closely the hypothesis fits the data for a given choice of $\boldsymbol \theta$.

Computer generated contemporary art

Just a quick selection of images randomly-generated by the following 10 lines or so of Python.