Which boundary conditions lead to existence and uniqueness of a second order PDE by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16www.cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap6.pdf 6.1 "Classification of PDE's" clarifies which boundary conditions are needed for existence and uniqueness of each type of second order of PDE:
Ciro Santilli is very fond of this result: the beauty of mathematics.
How can so much complexity come out from so few rules?
How can the proof be so long (thousands of papers)?? Surprise!!
And to top if all off, the awesomely named monster group could have a relationship with string theory via the monstrous moonshine?
The classification contains:
- cyclic groups: infinitely many, one for each prime order. Non-prime orders are not simple. These are the only Abelian ones.
- alternating groups of order 4 or greater: infinitely many
- groups of Lie type: a contains several infinite families
- sporadic groups: 26 or 27 of them depending on definitions
Simple Groups - Abstract Algebra by Socratica (2018)
Source. Good quick overview.2023-07: one of the sides broke near center, rebuying.
2021-07: wire half broke near connector, only works in some positions. The funny thing is: only voices seem to be blocked out! Rebuying.
2021-06: a small bottom piece of the left earpiece broke. Wire seems find, that is like a little extension to protect wire. Let's see for how long.
2020-20: wires at one of ears broke, not sure how.
Look they looked exactly like: "Skullcandy Smokin' Buds 2 In-Ear Audio Earbud Headphones with In-Line Microphone - Black".
Re-buying that one 10 pounds:
Tech specs:
Connection Type: 3.5mm AUX CableImpedence: 32 ohmsDriver Diameter: 9mmSound Pressure Level: 95 dB (1mW/500Hz)Frequency Response: 20kHz - 20HzHeadphone Type: In-Ear
In degrees Celsius:
- 25+
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- 0-10:
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- "Karrimor X Lite Run Black Headband". Head a bit cold on lower range.
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This is how electronic circuits are normally prototyped!
Once you validate them like this, the next step is usually to move on to printed circuit boards for more reliable production setups.
Breadboards are a thing of beauty and wonder.
Point-to-point constructions on woden boards
. Source. Predecessors to breadboards from where the name came. A thing of beauty, so vintage. You could actually write stuff on those with a pencil!Breadboards - Trash or Treasure? by Keysight (2020)
Source. 
How to build it: math.stackexchange.com/questions/3137319/how-in-general-does-one-construct-a-cycle-graph-for-a-group/3162746#3162746 good answer with ASCII art. You basically just pick each element, and repeatedly apply it, and remove any path that has a longer version.
Immediately gives the generating set of a group by looking at elements adjacent to the origin, and more generally the order of each element.
TODO uniqueness: can two different groups have the same cycle graph? It does not seem to tell us how every element interact with every other element, only with itself. This is in contrast with the Cayley graph, which more accurately describes group structure (but does not give the order of elements as directly), so feels like it won't be unique.
Stable isotope.
where:
- : matrix in the old basis
- : matrix in the new basis
- : change of basis matrix
Subtle is the Lord by Abraham Pais (1982) page 85:so it is quite cool to see that organic chemistry is one of the things that pushed atomic theory forward. Because when you start to observe that isomers has different characteristics, despite identical proportions of atoms, this is really hard to explain without talking about the relative positions of the atoms within molecules!
However, it became increasingly difficult in chemical circles to deny the reality of molecules after 1874, the year in which Jacobus Henricus van't Hoff and Joseph Achille Le Bel independently explained the isomerism of certain organic substances in terms of stereochemical properties of carbon compounds.
There are unlisted articles, also show them or only show them.