dressedsalad

[dressedsalad]

going insane im using bayes' theorem to estimate the likelihood of someone being gay & into me given their behaviour towards me but i'm beginning to need more standardized data

[dressedsalad]

this email could have been a battle to the death

[dressedsalad]

also for finite fields there's a similar set of matrices for all n, since for a field of cardinality p^k you can just see the field of cardinality p^(k*n) as a vector space over our original field, which gives us a division algebra so it gives us the solution

why does the amount of linear algebra any graduate math course expects you to know greatly exceed the amount of linear algebra any math course will ever teach you

[dressedsalad]

Oh ye i think this should work, checking the axioms for an algebra is obvious, to show divisibility we need to solve uniquely a=b x for any non null a and b, but this is like ezy because the map {b}xR^n->R^n is an isomorphism because b as a matrix has determinant 0, so there is exactly one element x which works, and the other way round works as well cuz R^n x{ b}->R^n is also an isomorphism, since no matrix sends b to 0. (To reiterate better, i'm identifying the span of the n matrices assumed to exist w the first copy of R^n, while the 2nd copy of R^n is just the set of vectors they act upon, so i get a map R^nxR^n->R^n by applying a matrix on the left to an element on the right) So this problem is exactly asking for what n there is an n-dimensional division algebra over R, so the answer do be 1,2,4,8 in virtue of a well known result (albeit hard to prove)

why does the amount of linear algebra any graduate math course expects you to know greatly exceed the amount of linear algebra any math course will ever teach you

[dressedsalad]

oh shee ye i think you're right, as long as it's a division algebra it works fine, in fact, maybe the map R^n xR^n into R^n can just make R^n into a non associative division algebra directly? It's non associative in general, it would be a division algebra by hypothesis tho i think maybe possibly idk

why does the amount of linear algebra any graduate math course expects you to know greatly exceed the amount of linear algebra any math course will ever teach you

[dressedsalad]

ok like you can't expand to 2^n like i thought, unfortunately if you take the determinant you get a polynomial on n variables which is greater *or equal* to zero and it indeed does "or equal" 0 at times. The last thingy should work, given how you get a map from P^n-1 x P^n-1 to P^n-1, you look at the cohomology rings over Z/2Z and you get that n has to be a power of two so that's nice. for n=4 a 4x4 matrix representation of the quaternions should work instead of my construction (wikipedia has one which seems to work) if the 8 case can be solved somehow through octonions it'd be great, not sure how since non associative algebras can't be really like, represented w real matrices. I've seen ppl mention k-theory, which i do not know, but regardless it'd be nice if it was possible to somehow get a division algebra from the existence of those n-matrices, so that the result could be at least come from a more well known result. Still gotta think how to get a n=8 solution to work if there is one, or how to quotient stuff the right way to get another cool topological map.

why does the amount of linear algebra any graduate math course expects you to know greatly exceed the amount of linear algebra any math course will ever teach you

[dressedsalad]

this shouuuld just be asking what n dimensional subspaces of the space of nxn matrices intersect trivially the hypersurface det(X)=0, for complex matrices it should like probably basically always do that (except for boring dimensions) , so rip, for odd dimensions and for matrices over R, the matrices X and -X will both be in a given n dimensional subspace, and taking a continuous path from X to -X, not crossing 0, since det(X)=-det(-X), eventually you'll end up w a matrix in your subspace w 0 determinant so no odds. Some candidates could be the dimensions 2^n, by starting in the 2x2 case w kaiasky's example and iteratively taking a matrix M in our choise of 2^n matrices and making the bigger matrices [M 0] [0 M] and [0 M][-M 0]. I suspect the fact that this would create a map from R^n x R^n->R^n which restricts to an isomorphism in {x}xR^n and in R^n x{x} is enough through alg topology magic to say the dimension should be 2^n, but i'll check tmrw i desperately need sleep rn

why does the amount of linear algebra any graduate math course expects you to know greatly exceed the amount of linear algebra any math course will ever teach you

[dressedsalad]

Send this to all your favorite moots and pass the pumpkin round! KEEP THE PUMPKIN TRAIN GOING 🎃🖤🎃🖤🎃

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHHHHHHH

[dressedsalad]

glorious, wonderful, showstopping, breathtaking etc etc

Yet another scrunkly,

Gonna make two matching ones for Christmas

[dressedsalad]

*having fantasies of starting a band and slowly falling in love with a random guy i saw on the street i didn't even talk to and thinking about writing the songs for said band and fixing up the logistics of everything involved and imagining a would-be day living with them and thinking about splitting the band as we grow older and dying together* huh maybe i am somewhat gay

[dressedsalad]

Bugs you

[dressedsalad]

AMV of the brown spider from minuscule

[dressedsalad]

Plato sweats bullets and shits himself once he realizes the beautiful nature that surrounds him is nothing more than shadows of statues much more complex than he's used to, and is all contained in a second bigger cave he wasn't aware of, which, in order to escape, requires defeating all the previous statues through increasingly difficult boss fights

[dressedsalad]

big fan of how if you look up any "generic enough to be somewhat popular" tag and sort by recent in the first 2 or 3 pages you'll almost always find a porn bot which spams random tags on a post consisting of a link to some shady ass website w a sexually charged pic which is the bot's 1 of like 2 or 3 posts, with no description and no reblogs but with a bunch of likes exclusively by other identical bots who do nothing but like each other's posts.

tumblr's strongest community, they support each other all the way through

[dressedsalad]

could you elaborate on how you found said homeomorphism? To me it seems like Y_1 is a closed subset of Y_ω that is not compact, hence Y_ω can't be compact.

Start with X_0 = R. To generate X_1, for each pair of points a, b in X_0 glue on a copy of [0, 1] connecting them. To generate X_{n+1}, take X_n and for each a, b in X_n glue on a copy of [0, 1] connecting them. Let X_ω be the colimit of this construction.

Is X_ω anything interesting?

Start with Y_0 = [0, 1]. To generate Y_1, for each pair of points a, b in Y_0 glue on a copy of [0, 1] connecting them. To generate Y_{n+1}, take Y_n and for each a, b in Y_n glue on a copy of Y_n connecting them (gluing in the obvious way). Let Y_ω be the colimit of this construction.

Is Y_ω anything interesting?

[dressedsalad]

as a mathematician i will always call maths the "purest" most abstract subject, as displayed by the famous xkcd comic, but i have to admit it's kind of a silly ranking criteria. Like fuck, biology sure is the most biological subject.

[dressedsalad]

hardcore reaction channel watching videos that are extremely challenging to react to