egg|nomz|egg changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> egg|nomz|egg: generally if your eyes are dewing over, that's not the weather. | <ferram4> I shall beat my problems to death with an engineer.
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<kmath>
<✔planet4589> Then, Ryugu's gravity will slowly accelerate Hayabusa-2 towards it until 2 days from now it will be falling at 5 c… https://t.co/K59bngcUXj
<kmath>
<whitequark> today I had a dream, a dream about food. ⏎ not just regular food, a delicacy that can only be created during a natura… https://t.co/nWsFAQ3fob
<kmath>
<whitequark> sure, cats live longer after hysteroovaritomy, but they also usually become docile blobs of fat, so i would persona… https://t.co/77ZDf6h64a
<kmath>
<whitequark> to be clear, this isn't a metaphor. i'm not that good at metaphors. i tried it (hope was a bit like chocolate, serv… https://t.co/K6YpYDl8d5
<egg|zzz|egg>
... I should try cooking that
<whitequark>
fucking impossible
<whitequark>
that star system was wild too
<whitequark>
actually like 2/3 of the dream was dedicated to the physics of it
<whitequark>
i think the closest i come to describing it is that the STAR had a day and a night half
<egg|zzz|egg>
Ꙩ_ꙩ
<whitequark>
no that isn't even the weirdest part
<whitequark>
so there was a planet that we were in orbit around that got deserted by its civilization long ago because it eventually got stuck on the night side of the star
<whitequark>
narrowly but enough to make it very cold there
<whitequark>
the planet had three satellites that co-orbited around the star, sort of like them being in lagrange points but all the points were close around the "host" planet
<whitequark>
the first satellite was always on the night side, heated from the inside by decaying radioisotopes, fertile but, well, without future
<whitequark>
the second was exactly between the night and the day side of the star, and it was rotating, and that was the world without sorrow
<whitequark>
the third was on the day side of the star, hot as hell and to make things worse, volcanic
<whitequark>
also they were all flat.
<whitequark>
(no, I don't know. radius under Roche limit? maybe?)
<egg|zzz|egg>
ꙮ_ꙮ
<whitequark>
anyway, the natural disaster was called "flat typhoon" and it was kind of like shadow storms in the Chronicles of Amber
<whitequark>
what it did is effectively swapped the places that all three satellites were placed in, exposing the advanced but stagnant civilizations on all three to the conditions their neighbor was facing
<whitequark>
they were advanced enough to not be destroyed immediately, and their struggle created the finest achievement ever known in that star system, but they had no chance, for they were stagnant for too long
<whitequark>
as for the "not embeddable in 𝐑⁴ part, I can't explain it through analogies for obvious reasons, and I can't explain it through math because I don't know enough math
<whitequark>
the system had some sort of symmetry that kept it rotating on a stable orbit, but the symmetry was decidedly not of kind that happens in euclidean space
<whitequark>
i'm pretty sure i observed at *least* the projection of 4-space onto 3-space
<whitequark>
that's the only way i can make any sense of it at all
<whitequark>
the anger sausages had eyes and the anger soup had spaghetti in it in the shape of span-4 routing wires in the iCE40 FPGA fabric
<whitequark>
unless you can get spaghetti to be flexible but still stay exactly parallel to one another in each pair with just one crossing towards the short side you can't cook this, egg
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<egg|zzz|egg>
so this general kind of fanciness should be feasible I think but otoh I don't know what the span-4 routing wires in the iCE40 FPGA fabric look like
<egg|zzz|egg>
yeah that sounds tedious but within the realm of possibility
<whitequark>
what about the "synesthetic flashbacks" part
<egg|zzz|egg>
yeah that seems harder :-p
<egg|zzz|egg>
whitequark: I made the pasta part of this thing http://forums.egullet.org/uploads/1204133002/gallery_8158_5665_68124.jpg once, it required spending half a day combing spaghetti covered in some sort of sauce, but it shouldn't be much worse to route them and have them cross over following that diagram
<pthag>
what is that
<egg|zzz|egg>
Jean François Piège's take on carbonara
<pthag>
well i can see the bacon hut
<whitequark>
wh
<pthag>
and i presume that's supercoiled spaghetti
<pthag>
what's the mustard cuboid
<egg|zzz|egg>
so I can tell you how the spaghetti block is made
<egg|zzz|egg>
you cast a block of cheese in a rectangle which forms the core; you comb spaghetti covered in a sauce into a sheet (making sure they are all parallel and straight, that's the extremely tedious part), and at the end you roll the spaghetti around the core
<whitequark>
can you cook spaghetti in already combed shape
<whitequark>
like hypothetically speaking uh
* whitequark
thinks about machining techniques
<egg|zzz|egg>
not without extremely fancy apparatus I think
<egg|zzz|egg>
oh you need to cool the spaghetti sheet so that the cooled sauce gives the sheet the requisite structural integrity to fold it over the center core
<whitequark>
you could embed spaghetti in a gel and heat the gel so they soak water from it while remaining in the same place
<whitequark>
call this "spaghetti fixturing"
<egg|zzz|egg>
I mean the sauce is that gel
<egg|zzz|egg>
hm
<egg|zzz|egg>
oh a heat resistant gel
<egg|zzz|egg>
yeah but then you need to remove it at some point
<whitequark>
yes! that's where sol-gel technique comes in
<whitequark>
make the gel stable only at certain pH values, for example
<egg|zzz|egg>
whitequark: but yeah if you could do this directly from straight spaghetti it would make life simpler, since those are easy to arrange in a straight sheet
<whitequark>
sauce tends to have non-neutral pH
<whitequark>
alternatively, you could just not boil them
<whitequark>
use something like sous vide
<whitequark>
if you don't boil them you don't run the risk of burning them if you use the absolute minimal amount of water
<egg|zzz|egg>
so sous vide won't really work here but maybe a rigid container at 90 degrees Celsius or so
<egg|zzz|egg>
whitequark: anyway, with the "spend a day combing them" technique, it would be easy to add the pair twists at the end that you describe
<egg|zzz|egg>
probably requires a firmer gel to hold it all in place, because they go a bit out-of-plane there
<whitequark>
would that really keep together in a soup?
<egg|zzz|egg>
ah, so there it gets interesting
<egg|zzz|egg>
you then *really* want a gel that holds itself at heat
<egg|zzz|egg>
which is not a problem, plenty of those
<egg|zzz|egg>
whitequark: a trickier question is how you put that noodle sheet in a soup at all, is it folded in some way, flat parallel to the surface, something else?
<egg|zzz|egg>
whitequark: also where are the sausages with respect to that circuit
<kmath>
<alonamit> Hilarious. Someone was tasked with creating a stock image for math cheating, so they plastered some random math for… https://t.co/Loo8OQBk5t
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<egg>
bofh: Εύρηκα how the moment of inertia works
* egg
is appropriately typing that from a bathtub
<bofh>
egg: ?
<egg>
bofh: theres actually zero difference between the 2nd moments tensor & the moment of inertia tensor.
* egg
bows head solemnly
* bofh
facepalms at self
<bofh>
*of course* :/
<egg>
that's nontrivial tho
<egg>
bofh: or, to be more explicit and less drilesque, let S be a symmetric and A antisymmetric, then {S, A} is antisymmetric and corresponds to (id*tr(S) - S) acting upon the hodged bivectors
<bofh>
Yeah, exactly.
<egg>
i.e. the change of sign and trace-time-identity business just comes from manipulating elements of so(3) as 3d vectors
<egg>
which means that taking a linear map that acts on vectors as a matrix A and making it act on bivectors in the naive way (still by matrix multiplication) is unnatural
<egg>
it *is* doable coordinate-free
<egg>
(because the inner product form is a thing and the trace too)
<egg>
but the natural way for it to act upon bivectors is by the anticommutator
<egg>
bofh: well, if A is not symmetric it's not the anticommutator, it's transpose(A) S + S A
<egg>
bofh: does that thing have a name
<bofh>
okay I *know* I've seen that particular Lie Bracket before in the context of classical Hamiltonian mechanics, sec.
<bofh>
nevermind I'm not sure, since we always have A symmetric in any case we've cared about. hm.
<bofh>
15:17:42 <@egg> which means that taking a linear map that acts on vectors as a matrix A and making it act on bivectors in the naive way (still by matrix multiplication) is unnatural
<bofh>
ohhhhhhhhhhhhh I think everything finally made sense right now.
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<hillexed>
hello everyone I'm here to steal your mathematical knowledge for personal gain before discovering that gain is rather hard to synthesize and giving up all but a twinkle of hope of attaining it
<APlayer>
Uh hi there hillexed. I've noticed you joined this channel as well as #KSPOfficial. Can we help you somehow?
<hillexed>
ah no that was a test message to make sure IRC was working
<APlayer>
In that case, I can confirm, it works. :-)
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<egg>
!wpn hillexed
* Qboid
gives hillexed a volatile [REDACTED]
<hillexed>
so basically a government contract
<bofh>
yeah more or less
<egg>
no that would be a const volatile [REDACTED]
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<UmbralRaptop>
!wpn hillexed && egg|z|egg
* Qboid
gives hillexed && egg|z|egg an optional turnstile
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<APlayer>
!wpn UmbralRaptop % egg|z|egg
* Qboid
gives UmbralRaptop % egg|z|egg a purring haloalkane
<hillexed>
wait how can a bivector be written as a skew-symmetric matrix
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<hillexed>
ah right writing the bivector as a skew-symmetric matrix is done in terms of the basis elements
<hillexed>
ah right writing the bivector as a skew-symmetric matrix is done in terms of the basis elements and I assume skew-symmetry comes from the anticommutivity of the wedge product?
<egg|laptop|egg>
well, because you can write the wedge as a difference of tensor products
<egg|laptop|egg>
v ∧ w = w ⊗ v - v ⊗ w
<hillexed>
(sorry about the spam I didn't realize this IRC client used grayed-out text to represent your own messages and not a lack of connectivity)
<egg|laptop|egg>
hillexed: another way to look at it is that V∧V is 𝖘𝖔(V)
<egg|laptop|egg>
which is why angular velocities, which are the derivatives of rotations, are bivectors
<hillexed>
oh wow nice
<egg|laptop|egg>
and then if you encode that in your type system you get the insane Principia libraries :-p
<hillexed>
ah, and that also makes sense because d of a 1-form should be a two-form
<egg|laptop|egg>
which 1-form?
<hillexed>
Rotations, I thought?
* egg
is confused, how are those 1-forms
<hillexed>
my intuition is that I'm thinking of rotations as points on S^2 and then reinterpreting those as vectors and I suspect my intuition is horribly wrong
<egg>
S² is off by a dimension for rotations
* egg
summons a Hopf fibration
<hillexed>
could you please unpack "which is why angular velocities, which are the derivatives of rotations, are bivectors" for me, then?
<hillexed>
thanks for being willing to explain
<egg|laptop|egg>
SO(3) is a Lie group
<egg|laptop|egg>
(or SO(V) I guess)
<egg|laptop|egg>
derivatives of paths in the Lie group lie in its Lie algebra, so in 𝖘𝖔(V)
<egg|laptop|egg>
so the derivative of an SO(V)-valued function of reals at any point is an element of 𝖘𝖔(V)
<egg|laptop|egg>
𝖘𝖔(V) is represented faithfully by antisymmetric matrices
<hillexed>
ah there's the problem: I haven't seen enough of Lie algebras to realize SO(V) != 𝖘𝖔(V)
<hillexed>
so the derivatives of SO(3) here, angular velocities, lie not in SO(3) but 𝖘𝖔(3), and each element of 𝖘𝖔(3) is a bivector?
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<egg|laptop|egg>
hillexed: in general, the derivatives of paths in a Lie group 𝐺 lie in its Lie algebra 𝖌. if 𝐺 is represented (in the representation theory sense of the term) as a matrix group, a subgroup of the group GL(n) of invertible n by n matrices for some n, 𝖌 will have a representation as a subalgebra of 𝖌𝖑(n), the algebra of n by n matrices (with the commutator as the Lie bracket)
<egg|laptop|egg>
now it happens that 𝖘𝖔(V) is canonically V∧V (using the inner product, but if you talk about SO(V) you must have an inner product)
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<egg>
!wpn whitequark
* Qboid
gives whitequark a tetracyclic pharmacology
<kmath>
<stephentyrone> @bofh453 @hikari_no_yume Also, yes, frequently much less precise with skanky libms.
<egg>
bofh: principia's Pow<2>(x) just does x*x though :-p
<bofh>
egg: yes but this is C, no templates.
<egg>
how can you live with a type system that's not Turing-complete,
<egg>
(yes I know Ada's is not)
<egg>
(there's a reason I don't use Ada anymore :-p)
<bofh>
(I thought it's b/c knowledge of Ada is fairly limited outside of, like, aerospace)
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<egg>
bofh: that's not a problem if I'm using a language for a personal project, is it
<egg>
it's more that it's a very good safe language for a language designed in 1983, but that has its limits
<egg>
memory pools are a poor way to manage memory except in very specialized cases, controlled types being done through an object-oriented approach is bad, generic instantiation being always explicit makes it clumsy, generics are not turing-complete, etc.
<egg>
bofh: also, aiui it's going out of favour even in aerospace now
<egg>
well, designed in the seventies really, with the first version being 1983
<egg>
also lots of features from Green didn't make it :-/
<kmath>
<eggleroy> @ManishEarth Interestingly, that is what the Green programming language (the proposal that led to Ada) did: functio… https://t.co/ZKFKENiTs4
<bofh>
okay, so I know eggstremely little about programming language theory. why is the distinction between functions and procedures meaningful?
<egg>
so functions in the green sense were pure
<egg>
i.e. no side effects
<egg>
which is nice, you can cache their results, reorder them at will, whatever
<bofh>
Okay, that makes sense, but how can you be certain that a function will be pure if you allow it to call other functions?
<egg>
bofh: because you allow it to call pure functions only :-p
<egg>
you don't allow it to set nonlocal variables, to mutate random pointees, etc.
<bofh>
okay, but from a functional perspective, how do you enforce that in the compiler?
<iximeow>
that seems exactly as difficult to enforce as 'const' functions
<egg>
bofh: assuming we are using a mix of Ada and Green terminology) within a function body, there can be no procedure calls, all access types used must be declared within the function, no tasking
<mlbaker>
egg: view the inner product as a map V->V* and apply the /\^k functor, presumably.
* egg
pokes mlbaker with the universal coded character set :-p
<egg>
yeah, I guess
<egg>
mlbaker: also see backlog for the realization that the inertia tensor is a bit silly and that it's the second moment tensor acting via the anticommutator that makes sense
<egg>
I like that I was recommended an algebra book to learn german and that I learned the word unmittelbar there :-p
<egg>
it means, of course, "by spending half an hour staring blankly at the page"
<egg>
very german, such concision
<bofh>
I mean that's accurate, every time you see the words "Proof: Obvious" it means you're spending the next hour staring at the page trying to figure out how it's obvious :P
<egg>
!wpn whitequark, коте, the котяchrome kitten, and the grey kitten
* Qboid
gives whitequark, коте, the котяchrome kitten, and the grey kitten a tagged compression
<kmath>
<BabelStone> CJK Unified Ideographs Supplement (32 code points) is squeezed between CJK Exts. B and C — the Platform 9¾ of CJK e… https://t.co/emEvkM1FOZ
<mlbaker>
something cute i recently learned, if Y->X is an n-sheeted ramified cover of Riemann surfaces, then line bundles on Y push down to rank r vector bundles on X.
<mlbaker>
rank n*
<mlbaker>
been reading hitchin's papers about moduli of stable bundles
<bofh>
okay, that's... actually a bit surprising. how's the proof of that work?
<mlbaker>
think of them as locally free sheaves
<mlbaker>
the intuition is just like
<mlbaker>
if x is a nonbranch point (has n distinct preimages)
<mlbaker>
then the fiber above x is just the direct sum of the n lines above its preimages
<bofh>
okay, I think I follow.
<mlbaker>
and then the branching behaviour manifests in the degree.
<mlbaker>
i.e. deg(E) in general won't be the same as deg(L)
<bofh>
ohhhhhhhhhh. okay now it makes sense.
<mlbaker>
and then the crazy shit is
<mlbaker>
the classical Riemann theta divisor is on J^{g-1}(X), right
<bofh>
Yeah that thing. It's kind of nasty to work with iirc.
<mlbaker>
the associated line bundle to that divisor has a 1-dim space of holomorphic sections
<bofh>
WAT
<mlbaker>
which are the 'order 1 theta functions'
<mlbaker>
(that's the classical story)
<bofh>
THIS MAKES WAY TOO MUCH SENSE AND I HATE THIS.
<mlbaker>
now the crazy shit is that if you instead look at the moduli space of stable rank r degree d vector bundles on X
<mlbaker>
you can basically imitate the def'n of the theta divisor there
<mlbaker>
and AGAIN, the associated line bundle has a 1-dim space of holomorphic sections.
<bofh>
and what are they in this case?
<mlbaker>
but in that case nobody knows how on earth to relate those to good old fashioned holomorphic functions of some sort.
<mlbaker>
but you have these Verlinde formulas
<mlbaker>
that the dimension of the space of sections of the kth power of that line bundle (which you might call 'kth order generalized theta functions')
<mlbaker>
is uh
<mlbaker>
well, it's freaky. there's a product involving sines and shit, rofl.
<kmath>
<whitequark> @eggleroy I am trying to restrain myself from writing a decent compiler frontend for it right now
<mlbaker>
p. 6, theorem 5
<bofh>
mlbaker: I have very hazy memories of those things from some night in undergrad, I recall it making no sense at the time (and I don't know if it will now :P)
<mlbaker>
yeah fair enough. this stuff seemed like absolute black magic to me just a few months ago
<SnoopJeDi>
TIL unmittelbar
<mlbaker>
actually my first brush with all these ideas was trying to read edward frenkel's langlands notes
<mlbaker>
and wondering what on earth higgs bundles were
<bofh>
oh those things. so basically you have a vector bundle E and a field operator φ that's a 1-form on End(E) with φ /\ φ = 0
<mlbaker>
highly recommend hitchin's paper "stable bundles and integrable systems" for that
<mlbaker>
yep
<bofh>
they're kind of generalizations of the Higgs field in condensed/particle
<mlbaker>
it's uh, bizarre how they kinda do exactly the right thing from a purely mathematical standpoint
* egg
pokes bofh with the Mathematical Operators block
<mlbaker>
like the spectral cover just pops out of nowhere
<bofh>
YEAH I came across that a year ago in a topics course and was like :???: this is completely unexpected
<mlbaker>
my advisor was like
<mlbaker>
"so little is known about this that people have developed a habit of getting really excited over publishing some trivialities on these objects"
<bofh>
ROFL
<whitequark>
18:43 * Qboid gives whitequark a tetracyclic pharmacology
<kmath>
<BabelStone> @ken_lunde @henryfhchan @eisoch Adding a new character is clearly the correct solution given that BOPOMOFO FINAL LE… https://t.co/UnpdplsPS4
<kmath>
<girlandkat> Our original version of this tweet noted the speed is close to “桜の花びらが舞い落ちる速度” or “the speed at which cherry blosso… https://t.co/xQropPli14
<egg>
(yes I refer to NBRs by their country code why :-p)
<bofh>
NBR?
<Qboid>
bofh: [NBR] => National Body Representative
<bofh>
Ahh.
<SnoopJeDi>
bofh, ;_;
<egg>
bofh: West is NBR for GB before ISO/IEC JTC 1/SC 2, Everson for IE
<egg>
(is it before SC 2 or SC 2/WG 2? now I'm confused)
<kmath>
<BabelStone> ... oddly the expert from UK abstained on encoding these emoji, whereas the experts from the Unicode Consortium and… https://t.co/dh1cgQEjuk
<egg>
whitequark: how do you feel about a Green frontend for kOS :-
<egg>
s/$/p/
<Qboid>
egg meant to say: whitequark: how do you feel about a Green frontend for kOS :-p
<Qboid>
U+1D598 MATHEMATICAL BOLD FRAKTUR SMALL S (𝖘)
<Qboid>
U+1D594 MATHEMATICAL BOLD FRAKTUR SMALL O (𝖔)
<mlbaker>
ahhh
<egg|nomz|egg>
bofh: it's amusingly commonplace here to have "let K be a field"
<egg|nomz|egg>
to the extent that seeing a general-porpoise field called F looks *weird* to me :-p
* UmbralRaptop
blinks. irssi is fine with unicode.
<egg>
UmbralRaptop: hm
<UmbralRaptop>
Is it running in cmd.exe, or something?
<mlbaker>
"Let k be a field, let k[i] = k[x]/<x^2+1>, consider \sum_{i=1}^k ..."
<egg|nomz|egg>
mlbaker: no, capital K
<egg|nomz|egg>
see, I'm in Switzerland
<mlbaker>
I've seen lowercase k as well a lot ><
<egg|nomz|egg>
that's evil
<mlbaker>
and then they use K/k for an extension field
<egg|nomz|egg>
D:
<egg|nomz|egg>
I've mostly seen L/K I think
<egg|nomz|egg>
or maybe primes
<egg|nomz|egg>
but using lowercase to name a field feels morally wrong
<mlbaker>
oh yeah, in an ANT context L/K is pretty common
<egg|nomz|egg>
ANT?
<mlbaker>
alg num thry
<egg|nomz|egg>
but yeah, if it's a Körper then you obviously call it K, and then when you've seen it called K everywhere and have learned K as your go-to letter you call it K in English
<egg|nomz|egg>
hence "let K be a field" :-p
<egg|nomz|egg>
My Coxeter group course started with "Sei W eine Gruppe. W wie Gruppe" ("Let W be a group; W for 'Group'")
<UmbralRaptop>
Ꙩ_ꙩ
<egg|nomz|egg>
:D
<egg>
why am i nomz
<egg>
i am actually zzz
egg is now known as egg|zzz|egg
egg|nomz|egg is now known as egg|z|egg
<egg|z|egg>
UmbralRaptop: W for Weyl of course, though the course never actually mentioned Weyl groups :-p
<kmath>
<rygorous> @stephentyrone "A dense block right in the middle of the matrix? Schurly you can't be serious!" - "I am serious, and don't call me Shirley."
<egg|zzz|egg>
UmbralRaptop: snek
<egg|zzz|egg>
"With respect to the inner product, exterior multiplication and the interior product" the <adjective> product situation is a mess
<UmbralRaptop>
fork(tongue)
<UmbralRaptop>
egg|zzz|egg: aaaaa
<egg|zzz|egg>
of course the exterior product is defined on top of the outer product,
<mlbaker>
every time i see the word 'schur', all i think of is "DET(GIANT MESS)/DET(OTHER MESS)"