it basically says that to prove a property Q(n,m) via "double induction" you first prove P(n)=Q(n,1) by induction (solid arrows) and then prove R(m)=Q(n,m) by induction (dashed arrows)

and i threw in some explanation about how ℕ² is well ordered with the lexicographic order and therefore it allows you to do that

that I read somewhere tho but I can see why this "double induction" strategy should be valid

I don't want to just throw in a cryptic procedure tbh i hate when grad school books do that

hs teachers: don't only cite the internet!

me, a phd student: btw my sources are MSE and MathOverflow

tbf I try to look up whatever I'm citing to make sure it makes sense

I thiiiiiiiink i will make a short appendix attaching the proofs and crediting but because the note is almost a page long and it's probably a level of technicality only a few people will bother with

@ZevenKorian Trick question are just mean. I include vague and marginal questions in that junkheap.

@AskChip i try to be as clear as possible because I'm covering a grad level topic but this book is aimed to last-year undergrads so of course my main source basically explains all of this (some 80+ pages and counting) in 10 pages

@ZevenKorian I always think proofs and exposition should have a clear DAG structure so it can be navigated by skipping bits you don't care about yet, then return to explore the details if you want to see how it works.

@ZevenKorian Hmm, I could swear I've seen that sidenote somewhere, but in English.

@ZevenKorian Now I remember, it was an introduction to a newer version of a math method that was beyond me, even in it's older form. I was so glazed over I can't even remember what the method was for but I think it was a description for an internal procedure for image processing.

@AskChip well, this is singular homology theory which is actually a method you use in image processing so it's not so far away c:

@ZevenKorian So it could be used in image recognition? That's something I was interested in but the math was too involved for me to grasp it except vaguely.

@AskChip homology theory basically designs methods to "scan" the shape of an object using algorithms, which you can then implement as programs (as they are fully automatic)

one example is the Mayer-Vietoris sequence, which is the basic method of computing homology (there are better methods but they may be harder to implement)

@AskChip for example, homology can distinguish between a hollow sphere (like a baloon) and a solid ball of concrete

one example I read about is using homology-assisted AI to detect tumors in an MRI scan

@AskChip a friend of mine actually did her thesis on implementing this method on java, i would joke to her about me doing the mathy details of her thesis

@ZevenKorian Java? OMG that's rich. I kinda liked Java but it scares me to use because it seems so structure resistant, kinda like many older flavors of Basic.

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