Open source LLM Updated 2025-07-19
Theories of Quantum Matter by Austen Lamacraft Quantum Hall Effect Appendix Updated 2025-07-16
Why do the electron and the proton have the same charge except for the opposite signs? Updated 2025-07-16
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.
Magnetic quantum number Updated 2025-07-16
Fixed quantum angular momentum in a given direction.
Can range between .
The z component of the quantum angular momentum is simply:so e.g. again for gallium:
- s-orbitals: necessarily have 0 z angular momentum
- p-orbitals: have either 0, or z angular momentum
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. is normally used by convention.
Mathematics course of the University of Oxford structure Updated 2025-07-16
Principal component analysis Updated 2025-07-16
is a hyperparameter, and 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 dimensional hyperplane. PCA is an algorithm for choosing this hyperplane and the coordinate system within this hyperplane.
The hyperplane choice is done as follows:
- the hyperplane will have origin at the mean point
- the first axis is picked along the direction of greatest variance, i.e. where points are the most spread out.Intuitively, if we pick an axis of small variation, that would be bad, because all the points are very close to one another on that axis, so it doesn't contain as much information that helps us differentiate the points.
- then we pick a second axis, orthogonal to the first one, and on the direction of second largest variance
- and so on until orthogonal axes are taken
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.
- flour
- dry codfish
- olive oil
- sausage
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.
Project Zomboid Updated 2025-07-16
This game is quite detailed: www.youtube.com/watch?v=w4Jmqp8a_bU
D-Wave Systems Updated 2025-07-16
Ethidium bromide Updated 2025-07-16
OsmAnd Updated 2025-07-16
The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns Updated 2025-07-16
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.
Vector graphics Updated 2025-07-16
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