The Einstein summation convention works will with partial derivatives and it is widely used in particle physics.

In particular, the divergence and the Laplacian can be succinctly expressed in this notation:

In order to express partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variables with indices such as $x_0$, $x_1$, $x_2$, ${\partial }_0$, ${\partial }_1$ and ${\partial }_2$ instead of the usual letters $x$, $y$ and $z$.

Can we do better than "wrong password implies random bytes"?

Can the last disk access times be checked via forensic methods?

By Fred Sanger's group.

Published as: sequence and organization of the human mitochondrial genome by Sanger et al. (1981).

The matrix ring of degree n $M_n$ is the set of all n-by-n square matrices together with the usual vector space and matrix multiplication operations.

This set forms a ring.

Related terminology:

Argued at Power, Sex, Suicide by Nick Lane (2006) page 212.

Basically, energy supply has to be modulated rather quickly, because we spend a lot sometimes, and very little other times.

Even not turning it off quickly enough is a problem, as it starts to generate free radicals which fuck you up.

If control came from the nucleus, it has no way to address different mitochondria. But it might be that only one of the mitochondria needs the change. If the nucleus tells all mitochondria to stop producing when only one is full, the others are going to say: "nope, I'm not full, continue producing!" and the one that need to stop will have its signal overridden by the others.

There are several choices of electromagnetic four-potential that lead to the same physics.

E.g. thinking about the electric potential alone, you could set the zero anywhere, and everything would remain be the same.

The Lorentz gauge is just one such choice. It is however a very popular one, because it is also manifestly Lorentz invariant.

For an example with context, have a look at E. Coli K-12 MG1655 and the second protein of the genome, e. Coli K-12 MG1655 gene thrA.

One notable application: cryptocurrency, see e.g. how Bitcoin works.

Used for example: