See: cirosantilli.com/china-dictatorship/orange-papers

In the classification of finite simple groups, groups of Lie type are a set of infinite families of simple lie groups. These are the other infinite families besides te cyclic groups and alternating groups.

A decent list at: en.wikipedia.org/wiki/List_of_finite_simple_groups, en.wikipedia.org/wiki/Group_of_Lie_type is just too unclear. The groups of Lie type can be subdivided into:

The first in this family discovered were a subset of the Chevalley groups $A_n(q)$ by Galois: $PSL(2,p)$, so it might be a good first one to try and understand what it looks like.

TODO understand intuitively why they are called of Lie type. Their names $A_n$, $B_n$ seem to correspond to the members of the classification of simple Lie groups which are also named like that.

But they are of course related to Lie groups, and as suggested at Video "Yang-Mills 1 by David Metzler (2011)" part 2, the continuity actually simplifies things.

"The Hot Troll Deviation" is the title of an episode from the popular TV show *The Big Bang Theory*, specifically season 4, episode 14. In this episode, the characters navigate various personal relationships and social dynamics. The storyline revolves around Raj's interest in a woman he meets online after he gets drunk and posts a risqué photo of himself, which leads to humorous situations. The episode explores themes of attraction and identity through its comedic lens, typical of the show's style.

A Ring can be seen as a generalization of a field where:

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ is not a ring because you can by addition reach the zero matrix.

TODO. Can't find it easily. Anyone?

This is closely linked to the Pauli exclusion principle.

What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.

Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.

As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.

Bibliography:

Ionizing radiation detectors are devices designed to measure and detect ionizing radiation, which includes particles and high-energy electromagnetic waves that have enough energy to remove tightly bound electrons from atoms, thereby ionizing them. Ionizing radiation includes alpha particles, beta particles, gamma rays, and X-rays. The working principle of these detectors typically involves the interaction of ionizing radiation with matter, which generates ion pairs (electron and positive ion) in a sensing medium.

Elliptic curve point addition is the group operation of an elliptic curve group, i.e. it is a function that takes two points of an elliptic curve as input, and returns a third point of the elliptic curve as its output, while obeying the group axioms.

The operation is defined e.g. at en.wikipedia.org/w/index.php?title=Elliptic_curve_point_multiplication&oldid=1168754060#Point_operations. For example, consider the most common case for two different points different. If the two points are given in coordinates:then the addition is defined in the general case as:with some slightly different definitions for point doubling $P+P$ and the identity point.

This definition relies only on operations that we know how to do on arbitrary fields:and it therefore works for elliptic curves defined over any field.

Just remember that:means:and that $y^{-1}$ always exists because it is the inverse element, which is guaranteed to exist for multiplication due to the group axioms it obeys.

The group function is usually called elliptic curve point addition, and repeated addition as done for DHKE is called elliptic curve point multiplication.

X-rays are a form of electromagnetic radiation, similar to visible light but with much higher energy and shorter wavelengths, typically in the range of 0.01 to 10 nanometers. They were discovered in 1895 by Wilhelm Conrad Röntgen and are widely used in various fields, most notably in medicine and science. In medical applications, X-rays are primarily used for imaging and diagnostic purposes. When X-rays pass through the body, they are absorbed at different rates by different tissues.

Made up mostly of calcium carbonate.

The Kennelly–Heaviside layer, also known as the E layer of the ionosphere, is a region in the Earth's upper atmosphere that is characterized by a high concentration of ionized particles. This layer is located approximately 30 to 100 kilometers (18 to 62 miles) above the Earth's surface and plays a significant role in radio wave propagation.

Grew tremendously since the 1990's, likely linked to the Internet.

www.cbpp.org/wealth-concentration-has-been-rising-toward-early-20th-century-levels-2 shows historical for top 1% and 0.5% from 1920 to 2010.

TODO why is it so hard to find a proper cumulative distribution function-like curve? OMG. This appears to be also called a Lorenz curve.

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:and in 2D:

The easy and less generic integral. The harder one is the Lebesgue integral.

The Fourier series of an $L^2$ function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.

The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).

Only proved in 1966, and known to be a hard result without any known simple proof.

This theorem of course implies that Fourier basis is complete for $L^2$, as it explicitly constructs a decomposition into the Fourier basis for every single function.

TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?

One of the many fourier inversion theorems.