Ciro Santilli @cirosantilli  40

Diffeomorphic to the 3 sphere.

The $U(1)$ unitary group is one very over-generalized way of looking at it :-)

The complex analogue of the special orthogonal group, i.e. the subgroup of the unitary group with determinant equals exactly 1 instead of an arbitrary complex number with absolute value equal 1 as is the case for the unitary group.

Most of the helium in the Earth's atmosphere comes from alpha decay, since helium is lighter than air and naturally escapes out out of the atmosphere.

Wiki mentions that alpha decay is well modelled as a quantum tunnelling event, see also Geiger-Nuttall law.

As a result of that law, alpha particles have relatively little energy variation around 5 MeV or a speed of about 5% of the speed of light for any element, because the energy is inversely exponentially proportional to half-life. This is because:

Uranium emits them, you can see their mass to charge ratio under magnetic field and so deduce that they are electrons.

Caused by weak interaction TODO why/how.

The emitted electron kinetic energy is random from zero to a maximum value. The rest goes into a neutrino. This is how the neutrino was first discovered/observed indirectly. This is well illustrated in a decay scheme such as Figure "caesium-137 decay scheme".

Most commonly known as a byproduct radioactive decay.

Their energy is very high compared example to more common radiation such as visible spectrum, and there is a neat reason for that: it's because the strong force that binds nuclei is strong so transitions lead to large energy changes.

A decay scheme such as Figure "caesium-137 decay scheme" illustrates well how gamma radiation happens as a byproduct of radioactive decay due to the existence of nuclear isomer.

Gamma rays are pretty cool as they give us insight into the energy levels/different configurations of the nucleus.

They have also been used as early sources of high energy particles for particle physics experiments before the development of particle accelerators, serving a similar purpose to cosmic rays in those early days.

But gamma rays they were more convenient in some cases because you could more easily manage them inside a laboratory rather than have to go climb some bloody mountain or a balloon.

The positron for example was first observed on cosmic rays, but better confirmed in gamma ray experiments by Carl David Anderson.

A protein that is a catalyst for some chemical reaction.

For an initial concrete example, consider e. Coli K-12 MG1655 gene thrA.

This simply connected is called the universal covering group.

E.g. in the case of $SO(3)$ and $SU(2)$, $SU(2)$ is simply connected, but $SO(3)$ is not.