Ciro Santilli @cirosantilli 37

As per en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965#Properties, unlike the Direct product, the semidirect product of two goups is neither unique, nor does it always exist, and there is no known algorithmic way way to tell if one exists or not.

This is because reaching the "output" of the semidirect produt of two groups requires extra non-obvious information that might not exist. This is because the semi-direct product is based on the product of group subsets. So you start with two small and completely independent groups, and it is not obvious how to join them up, i.e. how to define the group operation of the product group that is compatible with that of the two smaller input groups. Contrast this with the Direct product, where the composition is simple: just use the group operation of each group on either side.

Product of group subsets

So in other words, it is not a function like the Direct product. The semidiret product is therefore more like a property of three groups.

The semidirect product is more general than the direct product of groups when thinking about the group extension problem, because with the direct product of groups, both subgroups of the larger group are necessarily also normal (trivial projection group homomorphism on either side), while for the semidirect product, only one of them does.

Conversely, en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965 explains that if $G=N⋊H$, and besides the implied requirement that N is normal, H is also normal, then $G=N\times H$.

Smallest example: $D_6=C_3⋊C_2$ where $D$ is a dihedral group and $C$ are cyclic groups. $C_3$ (the rotation) is a normal subgroup of $D_6$, but $C_2$ (the flip) is not.

Note that with the Direct product instead we get $C_6$ and not $D_6$, i.e. $C_3\times C_2=C_6$ as per the direct product of two cyclic groups of coprime order is another cyclic group.

TODO:

Bibliography: math.stackexchange.com/questions/1726939/is-this-intuition-for-the-semidirect-product-of-groups-correct

Only certain battery voltages exist, because this voltage depends intrinsically on the battery's chemical composition.

learn.sparkfun.com/tutorials/battery-technologies/all (CC BY-SA) has a very good summary list, reordered from lowest to highest voltage:

Bibliography:

Like a heat equation but for functions without time dependence, space-only.

TODO confirm: does the solution of the heat equation always converge to the solution of the Laplace equation as time tends to infinity?

In one dimension, the Laplace equation is boring as it is just a straight line since the second derivative must be 0. That also matches our intuition of the limit solution of the heat equation.

Uniqueness: Uniqueness theorem for Poisson's equation.

There are actually two possible definitions for the DFT:

The $1/\sqrt{N}$ is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.

Most commonly, boundary conditions such as the Dirichlet boundary condition are taken to be fixed values in time.

But it also makes sense to think about cases where those values vary in time.

Some bibliography:

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?"

math.stackexchange.com/questions/2382011/computational-complexity-of-modular-exponentiation-from-rosens-discrete-mathem mentions:can be calculated in:Remember that $log(m)$ and $log(n)$ are the lengths in bits of $m$ and $n$, so in terms of the length in bits $M$ and $N$ we'd get:

Data that is inscribed in a blockchain as a way to perpetuate the data, rather than to follow the main intended purpose of the given blockchain, e.g. ASCII art instead of financial transactions on the Bitcoin blockchain.

A catalogue for Bitcoin can be found at: Section "Cool data embedded in the Bitcoin blockchain".

More info at: docs.ourbigbook.com#ourbigbook-web-topics

The most important ones are:

These are the key mathematical ideas to understand!!

There are actually a few formulations out there. By far the dominant one as of 2020 has been the Schrödinger picture, which contrasts notably with the Heisenberg picture.

Another well known one is the de Broglie-Bohm theory, which is deterministic, but non-local.