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.
<UmbralRaptop>
By the tome you have a tenure track job, haven't you gotten past all the really high risk areas?
<UmbralRaptop>
*time
* oeuf
scritches UmbralRaptop
<UmbralRaptop>
(I mean, if you saw me sadposting in wq-nsfw, you know that I'm touch starved)
* UmbralRaptop
pets oeuf
<Ellied>
!wpn oeuf
* Qboid
gives oeuf a thermionic vorpal gas
<Ellied>
yikes
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<UmbralRaptop>
cause of death: decapitation via cathode rays
<iximeow>
!wpn UmbralRaptop
* Qboid
gives UmbralRaptop a reticle with a kindle attachment
<iximeow>
ah perfect for the stabs
<UmbralRaptop>
!wpn iximeow
* Qboid
gives iximeow a finite cormorant
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<oeuf>
!wpn bofh, Ellied, iximeow, UmbralRaptop
* Qboid
gives bofh, Ellied, iximeow, UmbralRaptop a global tensor
<oeuf>
Ellied: how is the ellying
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<oeuf>
!wpn whitequark in absentia
* Qboid
gives whitequark in absentia a transuranic plasma
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<oeuf>
UmbralRaptop: what happened to your cuneiform displayname!
<oeuf>
!u ️
<Qboid>
U+FE0F VARIATION SELECTOR-16 (◌️)
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<UmbralRaptop>
oeuf: slow realization that I should make myself look more professional >_<
<egg|cell|egg>
UmbralRaptor: meh
<egg|cell|egg>
UmbralRaptor: it's Twitter not your cv
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<oeuf>
UmbralRaptop: atlas did have ~Long Long Cat~ as his display name for a while :-p
* oeuf
pets UmbralRaptop
<oeuf>
!wpn UmbralRaptop
* Qboid
gives UmbralRaptop a holomorphic baryon which strongly resembles a vortex
<UmbralRaptop>
oeuf: Atlas has tenure, though, right?
<UmbralRaptop>
Er, wait. I'm thinking of the @johnreghr
<oeuf>
UmbralRaptop: yeah, atlas works for the fruit company
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* raptop
apparently ran out of disk space on that system where I had all the compute. <_<
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BPlayer is now known as PlayerDisguisedAsUser
PlayerDisguisedAsUser is now known as BPlayer
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<oeuf>
UmbralRaptop: meow
<oeuf>
!wpn UmbralRaptop
* Qboid
gives UmbralRaptop a zsh-compatible |raw⟩
<raptop>
!wpn oeuf
* Qboid
gives oeuf a spectral polyether
<oeuf>
!pet raptop
* Qboid
pets raptop
* raptop
stares at things
<oeuf>
!meow raptop
* Qboid
meows at raptop
<raptop>
!pet oeuf
* Qboid
pets oeuf
<BPlayer>
So, I've been thinking of stuff lately. If you have a function, say, f(x) = a / (x - b)^c + d, you need four distinct x -> f(x) pairs to determine the function, right?
<BPlayer>
In this case it's four, that is
* raptop
stares at that c with concern
<BPlayer>
...alright, let's remove that c
<BPlayer>
a / (x - b) + d
<raptop>
(a / (x -b)) + d?
<raptop>
just to confirm
<BPlayer>
Yes
<BPlayer>
Though, AFAICT, it should not matter for my question
<raptop>
Okay, I think I understand it a bit better? As in a affects the equation with vertical scaling, b with a horizontal offset, d with a vertical offset, c with the slope?
<raptop>
Yeah, 3 pairs for the case without c seems right
<BPlayer>
Which is 3 variables and 3 pairs needed to find them
<BPlayer>
If you take 1 / (x - b) + d, you have 2 variables, 2 pairs
<BPlayer>
So /usually/, you need 1 pair per variable
<BPlayer>
However, obviously, a / (x - b + e) + d still needs only 3 pairs, though you have 4 variables. In this case it is obvious, but what would be a mathematically correct way to distinguish which variables need to be determined, and hence, for which variables you need such a "data pair"?
* BPlayer
looks at oeuf
<raptop>
Something about algebra. If this were a normal polynomial I'd feel more comfortable
<oeuf>
as raptop says if it's all polynomials it's algebra in some way (maybe algebraic geometry?); for arbitrarily strange functions it feels more like differential geometry?
<oeuf>
BOFH
<BPlayer>
...alright, let's say f(x) = ax^3 + bx^2 + cx + d. You have 4 variables and need 4 pairs. g(x) = x^3 + bx^2 + cx + d: 3 variables, 3 pairs. h(x) = n(ax^3 + bx^2 + cx + d): 5 variables, 4 pairs needed
* oeuf
casts "summon bofh"
<oeuf>
yes here it's just polynomials so it's just the dimension of your vector space
* raptop
sets up a light source and 25 grams of high quality chocolate
<oeuf>
otherwise i guess you have some fancy manifold but i'm principiaing right now
<oeuf>
raptop: good idea
<BPlayer>
I mean, in those cases it's relatively obvious, but I can see how you can disguise such an extra variable in such a way that it is not at all obvious
<BPlayer>
And mislead someone into thinking they need 5 pairs where, in fact, they only need 4. And there must be some algorithm to break it down?
<oeuf>
some algorithm? what am i, a computer scientist?
<BPlayer>
(I have no specific application or idea in mind, just wondering about a curiosity I found)
<raptop>
oeuf: computer science is actually a branch of mathematics, right?
<oeuf>
raptop: a boring one :-p
<BPlayer>
"Algorithm" is not necessarily used in conjunction with computers, though :-)
<oeuf>
for polynomials it's really algebraic geometry tbh
<raptop>
I mean, the 1/x, etc requires an infinite series of polynomials... >_>
<oeuf>
and iirc bofh has been poking at algebraic geometry
<oeuf>
how do i page bofh
<raptop>
oeuf: "it's not a science, and the computers are optional" (to paraphrase e_14159)
<oeuf>
hmm
<BPlayer>
raptop: Thank Mr. Taylor, not me :P
* raptop
pokes bofh
* raptop
pokes Taylor with a finite state machine
<e_14159>
raptop: Well, computer science is a *useful* branch of mathematics.
<e_14159>
AKA one that's not fun.
<BPlayer>
Anyway, I was mostly hoping that there is a name for the problem I described and that I may read up some articles on it
<BPlayer>
But apparently it is a nameless problem, so let's name it BProblem, after its discoverer. :P
<e_14159>
(Also, that quote isn't from me, but from some introductory computer science lecture)
<raptop>
e_14159: hey, now. Probability and statistics is fun. (Aside from how inpenetrable Bayesian stuff seems to be)
<BPlayer>
And egg is a cell, can't argue with that
<BPlayer>
An*
* BPlayer
can't type today
<bofh>
oeuf: I mean if it was a classical polynomial, it's just linear algebra (amounts to inverting a matrix).
<BPlayer>
I mean, the issue is that I know high school mathematics and no more, at least yet. So "inverting a matrix" sounds like black magic to me :P But I shall look it up one day
<egg|cell|egg>
Bofh: I feel like you can summon varieties in there
<bofh>
I'm not sure what's the simplest framework that will work with that one, but I know it can be computed via a Gröbner basis and will have minimal degree equal to the number of controllable degrees of freedom
<bofh>
like the nasty thing is entirely the fact that you have a 1/(x - b) that's throwing the naive application of the theory off
<bofh>
b/c it's a coefficient of degree -1 and that's < 0
<egg|cell|egg>
But in the initial example there was an exponentiation
<egg|cell|egg>
Can it be turned into a question of manifold dimension or something
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<bofh>
egg|cell|egg: the orignal eggsample is nastier, but still can be reduced to a Gröbner basis computation, so technically that *would* make it quality as algebraic geometry,
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<egg|cell|egg>
Bofh: for arbitrarily nasty functions can we summon diff. geometry
<bofh>
prolly, tho I'm unsure which incantations are involved.
<bofh>
Iä, Iä,
<bofh>
A͝ti̕y̨ah̴-Si͡n͜g̴e҉r͡?
<egg|cell|egg>
Hm
<egg|cell|egg>
Bofh: isn't it just compute Jacobian rank or something
<Qboid>
U+1D593 MATHEMATICAL BOLD FRAKTUR SMALL N (𝖓)
<Qboid>
U+1D58E MATHEMATICAL BOLD FRAKTUR SMALL I (𝖎)
<Qboid>
U+1D588 MATHEMATICAL BOLD FRAKTUR SMALL C (𝖈)
<Qboid>
U+1D58A MATHEMATICAL BOLD FRAKTUR SMALL E (𝖊)
<BPlayer>
Probably it matches "weird letters" with regular ones, and compares that to a dictionary. So when it found a potentially accidentally messed up equivalent of the dictionary word "nice", it "fixed" the Unicode
<egg|cell|egg>
It just nfkcs
<egg|cell|egg>
Or d considering Apple
<BPlayer>
Anyway, I am off for today. Thank you for looking into my question, even if I only understood "Bahnhof". ;-)
<BPlayer>
Good night!
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<oeuf>
bofh: how isn't BPlayer's question not just "what's the rank of the jacobian" actually
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<bofh>
oeuf: hrm, oh I guess this is still a polynomial and so the k'th order partials eventually terminate
<oeuf>
bofh: now im confused
<bofh>
oeuf: you can take a Jacobian and it makes sense
<bofh>
and so this is just the rank of the Jacobian
<oeuf>
the question is "given f : R^n+1 -> R, how many couples of reals (x,y) with y = f(v, x) determine v"
<bofh>
this is literally the "linear algebra" I alluded to "if the problem were nicer", i.e. lacked any 1/ax^n terms
<oeuf>
so for fixed x, y, we have some manifold y = f(v, x)
<oeuf>
and the question is the dimension of that manifold
<oeuf>
(assuming x,y in general position because it can obviously degenerate)
<bofh>
you don't even need to go that far, this is just a standard matrix, just pick your reals (x,y) to be the lagrange interpolant points
<bofh>
done
<oeuf>
bofh: but here you're making assumptions about f
<oeuf>
initially f was quite nasty, and you may want a nastier one
<bofh>
oeuf: unsurprisingly you can rapidly get nastiness if you throw out differentiability a.e., so I think that's a necessary criterion for eggsact interpolation from an amount of points less than the function's entire domain. now what's *sufficient* for, say, non-polynomials that still feel like they should be nice and admit low-degree interpolants, I have *no* idea.
<oeuf>
bofh: no i mean the question is just the implicit function theorem?
<bofh>
"just"
<oeuf>
i don't see why you bring interpolants in there
<oeuf>
bofh: e.g. i think figuring out that a cos^2 x + b cos 2x - a/2 actually has 1 dof is in scope of the original question
<oeuf>
sure you can do it in the special case of polynomials and it's all linear algebra
<bofh>
I mean yes, but that function is still differentiable once a.e. (it's C^{\infty}, even).
<oeuf>
sure
<bofh>
like I was trying to say differentiability a.e. is necessary, in which case implicit f'n theorem applies, but may be horrible to work with, and if you drop that condition, here be dragons?
<oeuf>
why is it horrible to work with
<oeuf>
the question looks like exactly the statement of the implicit function theorem?
<bofh>
erm, it is
<oeuf>
for each point (x, y) you constrain v by f(v, x) - y = 0, take grad_v f(v, x), etc.
<oeuf>
!tell BPlayer the implicit function theorem
<Qboid>
oeuf: I'll redirect this as soon as they are around.
<bofh>
oh right it gives you all the partials of g in your open set U via a matrix product
<bofh>
nevermind, it's not nasty, since you can use it to get a form of g that you can work with. :P
<oeuf>
not sure whence g here
<oeuf>
but okay
<oeuf>
bofh: seems like we both need a calculus 1 refresher >_>
<oeuf>
or is it 2
<oeuf>
whatever
<oeuf>
okay possibly 2
<oeuf>
bofh: admittedly i'm more of a discrete maths person, but i'd eggspect the physicist to remember some calculus :-p
<oeuf>
bofh: okay technically implicit function theorem tells you "are those points enough or too many"
<oeuf>
bofh: i.e. given points you look at linear dependence and codimension
<bofh>
oeuf: I mean I legitimately haven't used implicit f'n theorem in years :p (and the last time I used it was in the context of enumerative combinatorics, so,)
<oeuf>
bofh: if you want to know "how many points in general position are needed", it doesn't feel fundamentally different
<oeuf>
probably a bit trickier because you need to handwave "general position"
<bofh>
I mean if that's all you need to know, then yes, that suffices.
* oeuf
flails while screaming "in general position"
<bofh>
I mean, by magical wave of one's hand,
<oeuf>
!wpn bofh
* Qboid
gives bofh a tunneling javelin-like vibrator
<oeuf>
!wpn UmbralRaptop
* Qboid
gives UmbralRaptop an equinoctial telescope
* UmbralRaptop
points the telescope at the moon
* oeuf
EARTHMOOs at UmbralRaptop
* UmbralRaptop
FORTRANs at œuf?
* UmbralRaptop
🔪 a computer in the 10 GiB user account storage limit
<oeuf>
UmbralRaptop: store the rest in wq's library-- oh wait,
<UmbralRaptop>
Hah!
<UmbralRaptop>
œuf: anyway, most of the space is taken up by bzip2 compressed markov chains, so I'm not sure if it would be more or less coherent than a typical datasheet.