so you'd expect an equivariant multi-jet, which is not

and actually the space of all multi-jets is Not defined simply by taking direct sums?

ANYWAY, the thing is, this bothers me because $$S$$ is a finite set, so it is invariant under the action given by the symmeric group right?

i didn't know you can still turn latex into ascii lol

anyway for multi-germs you set the multi-jet, which is defined as the tuple of mono-germs at the points

for example, if $$S=\{x_1,x_2\}$$, we have
𝑗ᵏ₂𝑓=(𝑗ᵏ𝑓(𝑥₁),𝑗ᵏ𝑓(𝑥₂))

anyway then you have multi-germs, which is when you take the germ for a finite set $$S$$ instead of a point (you can think of it as having a bundle of mono-germs, since manifolds are usually assumed to be Hausdorff because life is too short to not assume everything is Hausdorff)

(See Golubitski and Guillemin's 1973 "Stable Mappings and Their Singularities" for details -- I think it's Chapter II, Section 2, Jet bundles)

so when you have a germ of smooth mapping between manifolds $$f\colon (N,x) \to P$$ you define the $$k$$-jet of $$f$$ at $$x$$ as the class of all $$(N,x) \to P$$ that have the same Taylor series expansion as $$f$$ up to order $$k$$, right?

(Taylor expansion can be computed taking charts in the source and target; while the coefficients are tied to the choice of charts, it can be proved that "having the same coefficients" is a consistent definition so long as you fix the charts)

i'm trying to find my post about legal prime numbers but i think i give up lol

we have the balanced hull, $$\tilde{D}$$, and we have the polybalanced hull, $$\tilde{\tilde{D}}$$, but may I interest you in

the hyperbalanced hull? $$\tilde{\tilde{\tilde{\tilde{D}}}}$$

from now on i will call sqrt[n]{x} arc^n

actually there's a lot to be asid about "no the idea is simple and straightforward but the calculations took me a whole page"

mfw last time i faced such an iff both were the multi-page one

Suppose you want to prove an iff statement with an easy direction and a tough as nails, multi-page direction. The easy one goes

so it turns out there Is a definition of integral mapping, like, in general

"Given an integral system I..."

boi it doesn't sound precisely enticing

is there a name for having a mapping $$f: N -> W$$ and "lifting it" on both source and target? so like, a name for getting F from f in this diagram: tikzcd.yichuanshen.de/#N4Igdg9

for the curious; yes, those T stand for tangent bundles

2-page proof, almost 3-page actually

a prof during the master's told us you can tell something is a theorem when the proof is long as hecc

and honestly, if it wasn't just two statements stapled together (because they're matching statements for you and your bestie) i would say that and call it a theorem lol

One of my favourite trivial facts about R-equivalence is that you can just divide by any unit

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