Viewing posts by christian

*This blog post was inspired by Holly Krieger's video for Numberphile.*

Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant $\pi$. One famous technique, attributed to the physicist James Jeans uses the number of letters in each word of the sentence:

The quadtree data structure is a convenient way to store the location of arbitrarily-distributed points in two-dimensional space. Quadtrees are often used in image processing and collision detection.

The UK's Ordnance Survey mapping agency now makes its 50 m resolution elevation data freely-available through its online OpenData download service. This article uses Python, NumPy and Matplotlib to process and visualize these data without using a specialized GIS library.

The equation for the temperature-dependence of the diffusion of hydrogen in tungsten may be written in Arrhenius form:
$$
k = A\exp\left(-\frac{E}{T}\right) \quad \Rightarrow \; \ln k = \ln A - \frac{E}{T},
$$
where the temperature, $T$, and activation energy, $E$, are expressed in eV and the pre-exponential Arrhenius parameter, $A$, and rate constant, $k$, take units of $\mathrm{m^2\,s^{-1}}$.
From the study of Frauenfelder [1] the parameters $A$ and $E$ may be associated with uncertainties as follows:
$$
\begin{align*}
A & = (4.1 \pm 0.5) \times 10^{-7}\;\mathrm{m^2\,s^{-1}}, \\
E &= 0.39 \pm 0.08 \;\mathrm{eV}.
\end{align*}
$$
These uncertainties can be propagated to the expression for $\ln k$:
$$
\sigma_{\ln k} \approx \sqrt{ \left( \frac{\sigma_A}{A} \right)^2 + \left( \frac{\sigma_E}{T} \right)^2 }.
$$
If we assume the uncertainty remains normally-distributed, Matplotlib's `imshow`

function can be used to illustrate the Arrhenius equation for this data.