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

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.

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

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 $\boldsymbol \theta$.

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