Ciro Santilli @cirosantilli 37

Given the view of the Standard Model where the electron and quarks are just completely separate matter fields, there is at first sight no clear theoretical requirement for that.

As mentioned e.g. at QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) chapter 1.6 "Hole theory", Dirac initially wanted to think of the holes in his hole theory as the protons, as a way to not have to postulate a new particle, the positron, and as a way to "explain" the proton in similar terms. Others however soon proposed arguments why the positron would need to have the same mass, and this idea had to be discarded.

Fixed quantum angular momentum in a given direction.

Can range between $\pm l$.

E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:

The z component of the quantum angular momentum is simply:so e.g. again for gallium:

Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. $z$ is normally used by convention.

Given a bunch of points in $n$ dimensions, PCA maps those points to a new $p$ dimensional space with $p\le n$.

$p$ is a hyperparameter, $p=1$ and $p=2$ are common choices when doing dataset exploration, as they can be easily visualized on a planar plot.

The mapping is done by projecting all points to a $p$ dimensional hyperplane. PCA is an algorithm for choosing this hyperplane and the coordinate system within this hyperplane.

The hyperplane choice is done as follows:

www.sartorius.com/en/knowledge/science-snippets/what-is-principal-component-analysis-pca-and-how-it-is-used-507186 provides an OK-ish example with a concrete context. In there, each point is a country, and the input data is the consumption of different kinds of foods per year, e.g.:so in this example, we would have input points in 4D.

The question is then: we want to be able to identify the country by what they eat.

Suppose that every country consumes the same amount of flour every year. Then, that number doesn't tell us much about which country each point represents (has the least variance), and the first PCA axes would basically never point anywhere near that direction.

Another cool thing is that PCA seems to automatically account for linear dependencies in the data, so it skips selecting highly correlated axes multiple times. For example, suppose that dry codfish and olive oil consumption are very high in Portugal and Spain, but very low in Germany and Poland. Therefore, the variation is very high in those two parameters, and contains a lot of information.

However, suppose that dry codfish consumption is also directly proportional to olive oil consumption. Because of this, it would be kind of wasteful if we selected:since the information about codfish already tells us the olive oil. PCA apparently recognizes this, and instead picks the first axis at a 45 degree angle to both dry codfish and olive oil, and then moves on to something else for the second axis.

We can see that much like the rest of machine learning, PCA can be seen as a form of compression.

This game is quite detailed: www.youtube.com/watch?v=w4Jmqp8a_bU

Vaporware?

Kind of works! Notably, has the amazing cycling database offline for you, if you fall within the 6 area downloads. It is worth supporting these people beyond the 6 free downloads however.

When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.

This is perhaps the best "default definition".

Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Smaller files, scalable image size, and editability. Why would you use anything else for programmatically generated images?!?!

The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies:where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

Initially there were mathematical reasons why people suspected that all boson needed to have 0 mass as is the case for photons a gluons, see Goldstone's theorem.

However, experiments showed that the W boson and the Z boson both has large non-zero masses.

So people started theorizing some hack that would fix up the equations, and they came up with the higgs mechanism.

As of 2020, basically means "liquid nitrogen temperature", which is much cheaper than liquid helium.

The dream of course being room temperature and pressure superconductor.

Haven't found the one yet:

Optional but really ideal:

The state of messaging is ridiculous as of 2020.

Why Wikipedia sucks: Section "Wikipedia".

Best languages:

The most important page of Wikipedia is undoubtedly: en.wikipedia.org/wiki/Wikipedia:Reliable_sources/Perennial_sources which lists the accepted and non accepted sources. Basically, the decision of what is true in this world.

Wikipedia is incredibly picky about copyright. E.g.: en.wikipedia.org/wiki/Wikipedia:Deletion_of_all_fair_use_images_of_living_people because "such portrait could be created". Yes, with a time machine, no problem! This does more harm than good... excessive!

Citing in Wikipedia is painful. Partly because of they have a billion different templates that you have to navigate. They should really have a system where you can easily reuse existing sources across articles! Section "How to use a single source multiple times in a Wikipedia article?"