jmblpati
[1] https://drops.dagstuhl.de/opus/volltexte/2021/14276/pdf/LIPI...
To understand why this even happens, let's say we want to compute the expression A*B + B*A for matrices A and B. One way we can do this is to compute the products A*B and B*A naively: two multiplications are needed. A trickier way to do this is to introduce a parameter x: we will instead compute A*A and (x*A + B/x)*(x*A + B/x) = x^2 A*A + (A*B + B*A) + x^-2 B*B. Thus (x*A + B/x)*(x*A + B/x) - x^2 A*A = (A*B + B*A) + x^-2 B*B: if x is large enough the second term vanishes and we can employ the rounding trick from before. Now this still needed two multiplications, but here one of our multiplications was A*A. If later we needed to compute A*C + C*A in the algorithm, we could then do that in only 1 additional matrix multiplication by repeating the trick. A more sophisticated version of this algorithm underlies all known approaches for matrix multiplication beyond w << 2.8.
However the block Krylov algorithm itself presented in this paper has a little bit more of a chance of being implementable than fast matrix multiplication (the matrix multipliciation is only used to solve small linear systems to deal with small eigendirections in the Krylov subspace). I am still skeptical that this is a truly practical algorithm due to its complexity, but unlike the case of generic FMM there is no obvious bottleneck.
On the other hand, in recent years Dan Spielman and collaborators have been working on fast implementations of Laplacian solvers: https://github.com/danspielman/Laplacians.jl I believe a lot of the fixed constants and combinatorial routines are changed from what is theoretically provable, but from screwing around with the code in the past it seems very fast in practice.
Let A have dimensions (a,b,c) and let B have dimensions (x,y,z). Assume A fits inside B.
We have (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc. This is the sum of the A's hypotenuse squared plus its surface area. The same holds for B.
Note that A's hypotenuse is at most that of B-- the hypotenuse of a is its longest axis, and it needs to fit in B somehow. Further, note that the surface area of A is less than that of B. To see this, consider the nesting of A inside B and realize that both boxes' interiors are convex sets. Imagine inflating A inside of B by taking the sets A_t consisting of all points within B that are within distance t of a point in A. It is not hard to see that this inflating operation can only increase the surface area of A, and since the maximum surface area we can get is that of B we have that A has smaller surface area than that of B. Thus,
(a+b+c)^2 = (A hypotenuse)^2 + (A surface area) <= (B hypotenuse)^2 + (B surface area) = (x+y+z)^2. The claim follows.
The only reason the wholesale disruption of hydrolases is not the thing that kills you with nerve agents is that acetylcholinesterase is particularly susceptible to organophosphates (wide open active site) and essential to nerve signal conduction. Thus although sarin (for example) will wreck most hydrolases in your body, this won't really matter when you'll be dead in a few minutes from your nervous system being unable to tell your lungs to breathe.
[0] https://en.wikipedia.org/wiki/IDFP
[1] https://en.wikipedia.org/wiki/Methoxy_arachidonyl_fluorophos...
The mechanism behind organophosphate poisoning lies entirely in the phosphate part of the molecule. Among the enzymes present in the human body are a large class known as serine hydrolases. These enzymes hydrolyze chemical bonds in a wide variety of compounds (such as sugars, proteins, and lipids), and their function critically depends on a nucleophillic serine amino acid in the enzyme's structure. This serine attacks the bond to be hydrolyzed, freeing one half of the victim molecule and leaving the other half attached to the enzyme. Then water comes in and breaks the bond between the serine and the second half to restore the enzyme and in sum hydrolyze the original bond.
Organophosphate nerve agents act by entering the active site of a serine hydrolase. They are attacked by the active site serine to release a counterion (which depends on the specific nerve agent) and a phosphate ester of the enzyme. However, this enzyme ester can only be hydrolyzed extremely slowly-- the enzyme cannot be restored to its original form, and the entire enzyme is nonfunctional.
Now, the released counterion could be many things (it just needs to be something that will leave if pushed out by the hydrolase serine). On the other hand, it should not be too eager to leave like something like chlorine would be: if it was the potential nerve agent would be attacked by things like water before getting to the hydrolase, wasting the agent. Thus, a balance needs to be struck between something easy to displace but not too easy to displace. This gives the classic functional groups present in most nerve agents and organophosphate pesticides: fluoride, cyanide, thiols, and phenolates.
By the way, fluorides and fluoride minerals are virtually impossible to regulate trade in: they have a large variety of legitimate uses from toothpaste to aluminum production, and fluorspar is a fairly common mineral in the world. What is usually regulated instead are the phosphorus precursors such as phosphorus trichloride and red phosphorus as well as some chlorinating agents like thionyl chloride.