raptop 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. | We can haz pdf
<egg|cell|egg>
mʲʷ
egg|zz|egg has joined #kspacademia
<mofh>
egg|zz|egg: today I was at a talk where the speaker (a very well-renowned physicist) used ℂ for a numerical constant in his slides. *THE* RELEVANT ONE FOR NUMERICAL CALCULATION, SO IT WAS ON HALF THE SLIDES. YES, ℂ. THERE WAS NO REASON TO USE THAT, HE ONLY USED LIKE 10 OTHER VARIABLES TOPS. AND BEFORE YOU ASK, IT WAS A REAL NUMBER TOO.
<mofh>
my eye is still twitching a bit.
<oeuf>
hah
<oeuf>
mofh: shall we use polynomials as an eggsample veggtor space to inflict linalg upon UmbralRaptor
<UmbralRaptor>
mofh: ∀∀∀∀∀∀∀∀∀
<oeuf>
it's easy to reason about while being sufficiently distinct from "just tuples of reals"
<oeuf>
UmbralRaptor: the set of polynomials in x with rational coefficients is a vector space
<oeuf>
UmbralRaptor: that is, you can add these polynomials (and you get a polynomial), you can multiply them by a rational
<mofh>
oeuf: LOL
<oeuf>
UmbralRaptor: like this: (the rational 17) * (the polynomial x^3 + 2) = (the polynomial 17x^3 + 34)
<oeuf>
UmbralRaptor: you can look up the set of axioms, but basically it has an additive structure, multiplication by a scalar, and the two play nice together (distributivity).
<oeuf>
UmbralRaptor: let's call this space of polynomials V
<oeuf>
UmbralRaptor: now let's consider the rational-valued functions of V that are linear
<oeuf>
UmbralRaptor: that is, functions λ : V → ℚ, such that for all polynomials v and w, and all rationals a, λ(0) = 0 and λ(av+w) = a λ(v) + λ(w)
<oeuf>
UmbralRaptor: eggsample!
<oeuf>
UmbralRaptor: λ(v) = 0 for all v is such a function. That was a boring example.
<oeuf>
UmbralRaptor: λ(v) = (the polynomial v evaluated at x=17) is also such a function
<oeuf>
UmbralRaptor: λ(v) = (the integral of the polynomial v, between x = 1 and x = 2) is also such a function
<oeuf>
UmbralRaptor: do these eggsamples make sense
<UmbralRaptor>
unsure about the second an third
<oeuf>
UmbralRaptor: so, for (the polynomial v evaluated at x=17), I will use the notation v(17), for concision
<oeuf>
UmbralRaptor: the statement is then (av+w)(17) = a v(17) + w(17)
<UmbralRaptor>
λ(v) = ∂v/∂x is obviously another, but I'm less clear on- ah
<oeuf>
UmbralRaptor: in other words, adding up and multiplying the polynomials as formal things that have an x in them, and then substituting, gives the same result as substituting, and then doing the calculation
<oeuf>
<UmbralRaptor> λ(v) = ∂v/∂x is obviously another <<< Achtung! we said rational-valued
<oeuf>
∂v/∂x is a polynomial, not a scalar
<oeuf>
∂v/∂x evaluated at 17 is a scalar
<UmbralRaptor>
δδ
<UmbralRaptor>
er
<oeuf>
and indeed, λ(v) = (∂v/∂x evaluated at 17) is another eggsample
<oeuf>
UmbralRaptor: can you think of other eggsamples of λs
<oeuf>
<UmbralRaptor> δδ << is that a smiley,
<UmbralRaptor>
phantom cat on the keyboard
<mofh>
not sure what smiley that would be,
<oeuf>
UmbralRaptor: okay so the following is true: you can add those λs, and you get something that still has that property
<UmbralRaptor>
other eggsamples would be λ(v) = 17*λ, though not 17+λ?
<oeuf>
yup eggsactly
<oeuf>
very eggsactly
<UmbralRaptor>
er, thouse last two λ should be v
<oeuf>
um
<UmbralRaptor>
those
<oeuf>
wait
<oeuf>
v is still a polynomial
<oeuf>
17*v is therefore a polynomial, not a scalar
<oeuf>
you want a λ that will turn x^3-x+2 into a rational number, not something with xs in it
<UmbralRaptor>
ah
e_14159 has quit [Ping timeout: 190 seconds]
<oeuf>
evaluation does that, definite integration does that, deriving and then evaluating the derivative does that
<UmbralRaptor>
so λ needs to always evaluate at some point or discard v?
<oeuf>
well
<oeuf>
it can do fancy things
<oeuf>
UmbralRaptor: see e.g. λ(v) = ∫ v(x) dx from 1 to 2
e_14159 has joined #kspacademia
<mlbaker>
i mean the space of rational polynomials of degree at most d is secretly just Q^{d+1} anyway
<oeuf>
mlbaker: yes i know, the point is to teach UmbralRaptor about vectors (and especially 1-forms, and then tensor products thereof) as abstract objects rather than intrinsically tuples :-p
<oeuf>
eventually i can point at some bases
<mlbaker>
ahh ok
<mlbaker>
over any field though V* is noncanonically isomorphic to V in the finite-dimensional case
<mlbaker>
so the functionals can't actually be too exotic
<oeuf>
mlbaker: see backlog, UmbralRaptor has essentially had differential geometry in GR + braket notation dropped onto him without any understanding of what V* even is, so the many notations to cope with V* and V have no motivation, leading to much confusion
<mlbaker>
oh god
<oeuf>
mlbaker: also the astute reader will notice that I have sneakily picked an infinite dimensional vector space here
<UmbralRaptor>
oeuf: well, for my porpoises, the definite integral involves evaluating a function at two points?
<oeuf>
UmbralRaptor: well, first you massage it a bit, and then you evaluate the massaged function at two points, yes
<oeuf>
UmbralRaptor: in fact since A(v) := ∫ v(x) dx (the indefinite integral this time) is a linear map from V to itself, we can express λ(v) = ∫ v(x) dx from 1 to 2 as λ(v) = (evaluation at 1 - evaluation at 2)A(v), but let's not get ahead of ourselves
<oeuf>
UmbralRaptor: other eggsample of such a λ: take the coefficient of x^17
<oeuf>
UmbralRaptor: when you add two polynomials, the x^17 terms add up, when you multiply a polynomial by a scalar, the x^17 term gets multiplied, so extracting that coefficient will be linear
<oeuf>
UmbralRaptor: okay now onto bestowing a structure upon those λs
<oeuf>
UmbralRaptor: let's call the space of those λs V*
<oeuf>
UmbralRaptor: then for λ1, λ2 in V*, we can define λ1+λ2 as (λ1+λ2)(v) := λ1(v) + λ2(v)
<oeuf>
and that newly-defined λ1+λ2 is in V* (that is, it's a linear rational-valued function of V)
<oeuf>
UmbralRaptor: given λ in V* and a rational b, we can also define bλ as (bλ)(v) := b * λ(v)
<oeuf>
UmbralRaptor: again bλ is in V* (left as an eggsercise to the reader, it's one of these things where you expand the definition and get what you want to prove immediately)
<oeuf>
UmbralRaptor: eggsample time again!
<oeuf>
UmbralRaptor: let λ1 be (evaluate at 1), λ2 (evaluate at 2)
<oeuf>
UmbralRaptor: then we have a new shiny function (λ1+λ2), which is (evaluate at 1, evaluate at 2, and add the two results)
<oeuf>
UmbralRaptor: in other words, (λ1+λ2)(v) = v(1) + v(2)
<oeuf>
UmbralRaptor: and that's an element of V* too
<oeuf>
UmbralRaptor: V* is called the dual space of V
<oeuf>
UmbralRaptor: its elements are called 1-forms, or linear forms
<oeuf>
UmbralRaptor: or if you're a physicist, depending on the kind of physics you're touching, they might be called covectors, or bras
<mlbaker>
or linear functionals (for historical reasons)
<oeuf>
meaningful hysterical raisins
<mlbaker>
actually even in this example
<mlbaker>
if you're willing to think about polynomials as functions, then an element of V* is a 'function of a function argument', hence functional
<oeuf>
yup
<UmbralRaptor>
functionals randomly popped up in Classical, and then vanished once we had extrema and Hamiltonians defined >_>
<mlbaker>
the True Way to think about polynomials is just as sequences of coefficients though
<oeuf>
mlbaker: and the derivative and integral are pretty nice eggsamples of linear maps too
<oeuf>
mlbaker: yes but which basis muahaha
<mlbaker>
the power basis
<oeuf>
meh, the monomial basis is boring
<mlbaker>
agreed but it's pretty canonical
<oeuf>
чебышёв ftw
<oeuf>
UmbralRaptor: note that, conceptually, the many-named animals in V* are fairly different-looking animals from the elements of V: "evaluate at 1" or "integrate from 1 to 2" or "derive thrice then evaluate at 17", vs "x^3-x+17" or "42"; even though, as mlbaker says, lurking beneath the surface are bases, any of which will make V* look like row vectors and V like column vectors (but we are going to pretend that they don't
<oeuf>
eggsist for a little bit longer)
<oeuf>
UmbralRaptor: of note: "42" is an element of V (a polynomial of degree 0 with constant term 42), it is *not* an element of V*: the function that returns 42 does not satisfy λ(0)=0
<UmbralRaptor>
silly piecewise functions can live in V*, I think?
<mlbaker>
'piecewise' isn't really a property of a function
<oeuf>
UmbralRaptor: at this point it is 3:16 so i need to zzz
<UmbralRaptor>
hm
<oeuf>
UmbralRaptor: try having some fun with the dual vector space in the meantime and tomorrow i'll introduce ⊗ (and hopefully manage a compromise between the rabbit hole of the universal property and the "pile of basis vectors" approach)
<oeuf>
unless mofh beats me to it ofc
<oeuf>
!wpn
* galois
gives oeuf a dysprosium boron atlatl
* oeuf
swats mofh with the atlatl
<oeuf>
UmbralRaptor: eggsample thing you can think about in the meantime: restricting yourself to polynomials of degree <= 3, can you express "integrate from 1 to 2" which is in V* as a linear combination of a bunch of "evaluate at <wherever>"
<oeuf>
!wpn mofh
* galois
gives mofh a partisan with a 🗡 attachment
<mlbaker>
wait do you need to get that complicated? they're just polynomials
<mlbaker>
integration is the unique linear map that sends x^n |-> x^{n+1}/(n+1)
<egg|zz|egg>
can anyone find King, R., Masters, R., Rizos, C., Stolz, A. (1987): Surveying with GPS. University of New SouthWales,Australia 1985,Verlag F. Dümmler Bonn 1987.
<egg|zz|egg>
yes i am good at sleeping why do you ask
egg|zz|egg has quit [Ping timeout: 183 seconds]
<mofh>
egg|zz|egg: moment
<mofh>
mlbaker: LOL
<mlbaker>
mofh: ?
<mofh>
egg|zz|egg: for some reason mofh was failing as a ping, i just fixed it.
<mofh>
mlbaker: defining integration that way
<mlbaker>
yeah, always choose C = 0
<egg|cell|egg>
Mofh: meow
<egg|cell|egg>
UmbralRaptor: yeah Simpson's rule is the right idea (but I said degree 3 so you need the Newton Cotes formula with 4 points, not 3)
<egg|cell|egg>
Oh it's also called Simpson's
egg|zz|egg has joined #kspacademia
<_whitenotifier-3d18>
[Principia] mariusvn forked the repository - https://git.io/fj32E
<egg|zz|egg>
!wpn mofh
* galois
gives mofh a zinc katyusha with a knife attachment
egg|zz|egg has quit [Ping timeout: 183 seconds]
* UmbralRaptor
🔪 sleep
egg|zz|egg has joined #kspacademia
egg|zz|egg has quit [Ping timeout: 183 seconds]
egg|cell|egg has quit [Ping timeout: 190 seconds]
egg|cell|egg has joined #kspacademia
egg|work|egg has quit [Quit: webchat.esper.net]
egg|laptop|egg has joined #kspacademia
<egg|laptop|egg>
!seen mofh
<galois>
egg|laptop|egg: I last saw mofh at 2019-04-23 - 02:41:25 in here, saying mlbaker: defining integration that way
egg|laptop|egg is now known as egg|work|egg
egg|cell|egg has quit [Ping timeout: 180 seconds]
egg|cell|egg has joined #kspacademia
egg|cell|egg has quit [Ping timeout: 202 seconds]
egg|cell|egg has joined #kspacademia
<mofh>
egg|cell|egg: meow
<egg|work|egg>
mofh: meow
egg|cell|egg has quit [Ping timeout: 202 seconds]
<egg|work|egg>
mofh: I was thinking of continuing the LinearRaptor thing with considering W⊗V* as a subspace of Hom(V, W), does that make sense
<egg|work|egg>
mofh: also should I introduce W⊗V* first and then talk about Hom(V,W) or the reverse,
egg|cell|egg has joined #kspacademia
<egg|work|egg>
mofh: also which is going to be least confusing between homomorphisms and automorphisms here, should i have a distinct space W
<mofh>
I've seen it done with starting from W⊗V* and going to Hom(V,W) and I think that makes more logical sense, but I do think this might be a YMMV thing.
<mofh>
mlbaker: ^
<mofh>
egg|work|egg: I think distinct space might be easier just to follow along
<egg|work|egg>
mofh: hm, yes, but I'm talking about polynomial spaces here, so the natural eggsamples (integration, derivation, etc.) tend to be automorphisms
<mofh>
True.
<egg|work|egg>
mofh: also in this case, V being infinite-dimensional, dual bases aren't bases, right
<mofh>
...on further thought it might actually just make sense to stick within polynomial spaces and the differentiation (derivation?) map in this conteggst
<egg|work|egg>
differivation
<mofh>
egg|work|egg: yes, the dual set of an infinite-dimensional V will not span its algebraic dual V^{#}, and things get fucky with regards to the continuous dual V^{*} (a dual basis exists, but it's not necessarily computable via the same way as dual bases for FDVSes)
<egg|work|egg>
mofh: I wrote the algebraic dual * in the above
<egg|work|egg>
mofh: I'm used to prime for the continuous dual
<egg|work|egg>
and * for the algebraic one
<mofh>
I'm used to * for the continuous dual and not ever caring about the algebraic dual
<egg|work|egg>
mofh: lol
<egg|work|egg>
mofh: but here we're just doing pure linalg so the continuous dual has not been introduced
<egg|work|egg>
mofh: let's not sidetrack UmbralRaptor into tempered distrubution
<egg|work|egg>
Die Wohltemperierte Verteilung
<mofh>
I mean at some point you'll want to introduce boundedness of operators (which I maintain is a bit silly of a term since what it actually means is Lipschitz-continuous, not bounded-by-constant).
<mofh>
But fair.
<egg|work|egg>
boundedness of operators << clearly a weird sun twitter account
<mofh>
Also more talks today so vanishing for a few hours
<mofh>
YES
<mofh>
OKAY I NEED TO MAKE THAT ONE LOL
<egg|work|egg>
mofh: but also not really, I mainly want to get to the point where the notation makes sense
<egg|work|egg>
both index notation and the bracket stuff
<mofh>
I mean if you're doing diffgeo I do think you run into at least *some* analysis out of necessity but also yeah, fair.
<egg|work|egg>
hence starting with talking about the algebraic dual and then tensor products as subspaces of the appropriate (multi)linear maps/forms
<egg|work|egg>
mofh: I'm not planning on doing diffgeo, hopefully the GR course does that
<kmath>
<Himmapaan> 'Princess Joveta sends her grateful thanks to her friend the great Khan for his kind gift of her new pet Velocirapt… https://t.co/8SjOMqPupF
Technicalfool has joined #kspacademia
<mlbaker>
Hom(V,W) is imo a more fundamental idea
<mlbaker>
in any category, discuss the objects, then the morphisms, then the extra shit
<mlbaker>
also (assume V and W fin-dim) defining tensor products via multilinear maps is fine as long as you behave yourself and call it by its rightful name V^* \otimes W^* rather than V \otimes W
<mlbaker>
and then mention the canonical V \cong V** and how that just gets implicitly milked literally everywhere
<egg|work|egg>
mlbaker well yes, implicitly dualizing would defeat the whole point here, which is to explain the idea underlying two notations that prominently distinguish the dual
<mlbaker>
and you should say something about the universal property of the tensor product
<mlbaker>
because like
<egg|work|egg>
I could, but I think that would confuse the diapsid
<mlbaker>
one thing i found super confusing back then is when i'm "allowed" to just be like "define a linear map V \otimes W -> Z by f(v \otimes w) = blah"
<mlbaker>
and the universal property is what answers that question
<egg|work|egg>
well yes, but you can go the other way around
<mlbaker>
?
<egg|work|egg>
identify the tensor product with a subspace of Hom, and define v⊗λ as the linear map which maps w to λ(w)v
<egg|work|egg>
and then define V⊗V* as the span of the pure tensors
<mlbaker>
i mean, the tensor product is all of Hom
<egg|work|egg>
well
<mlbaker>
in the finite dimensional case
<egg|work|egg>
in finite dimension yes
<egg|work|egg>
:-)
<mlbaker>
and for diff geo pretty much everything is finite dimensional
<egg|work|egg>
(I may have been that student in first year linear algebra who was always interjecting "you forgot to say 'assuming finite dimension'")
<egg|work|egg>
mlbaker: yes but in QM the spaces aren't so nice
<mlbaker>
oh god
<mlbaker>
I mean, isn't he just trying to learn this for GR?
<egg|work|egg>
both
<egg|work|egg>
confused by both QM and GR at the same time
<mlbaker>
QM is way more subtle than elementary riemannian geometry imo
<egg|work|egg>
with nobody telling him that bras and lower indices have something to do with each other and that V* exists
<mlbaker>
to do mathematically rigorously
<egg|work|egg>
yeah i'm not getting into tempered distributions as i told mofh
<egg|work|egg>
but at least if the formalism is there for matrix mechanics we'll have something
<mlbaker>
and the concept of a continuous dual only makes sense for a TVS or something
<egg|work|egg>
my goal here is neither to do a diffgeo course nor a QM course, merely to have some grounding in linalg that these courses are built upon but that UmbralRaptor is missing
<mlbaker>
yeah
<egg|work|egg>
(might get into the diffgeo if then the diffgeo part of his GR course proves stupid, but let's start easy)
<mlbaker>
so probably not much needs to be said about infinite dimensions aside from "the dual basis doesn't span if V is infinite-dimensional, and not all linear maps are finite rank"
<egg|work|egg>
yeah
<egg|work|egg>
but the space of polynomials is a nice eggsample here
<mlbaker>
btw, derivation and integration are not actually automorphisms
<mlbaker>
derivation has a kernel
<egg|work|egg>
s/auto/endo/
* egg|work|egg
slaps egg
<mlbaker>
ah
<UmbralRaptor>
mlbaker: TVS as in that GR replacement Bekenstein came up with?
<mlbaker>
TVS = topological vector space
<egg|work|egg>
topological veggtor spaces
<UmbralRaptor>
ah
<UmbralRaptor>
(well, it's a name)
<mlbaker>
finite dimensional thingies are all TVS in a unique way; the subtlety comes in infinite dimensions
<egg|work|egg>
mlbaker: with this eggsample you can limit the degree and get finite dimensional spaces, and you can trivially exhibit the oddities of the infinite-dimensional case that lurks in the background, it's a cute vector space
<mlbaker>
yeah
<mlbaker>
\bigoplus_{i=1}^\infty R
<mlbaker>
should contrast that with \prod ;)
<egg|work|egg>
well we could start talking about dimensionful physical quantities and express them as pure tensors in 1d but let's not go there
<mlbaker>
i remember we talked about the dimensionful thing a while back here
<egg|work|egg>
yeah
<egg|work|egg>
it is inconvenient to talk about because of the word dimension,
<egg|work|egg>
clearly we should call that the grandeur
<egg|work|egg>
and then dimension can retain its vector space meaning
* UmbralRaptor
assumes that this makes more sense in French?
<mlbaker>
(infinite dimensional) analysis vs infinite (dimensional analysis)
<egg|work|egg>
"grandeur physique" is how you would say "physical quantity"
<egg|work|egg>
mlbaker:aaaaaaaa
<egg|work|egg>
UmbralRaptor: physical greatness,
<UmbralRaptor>
hrm
<mlbaker>
also the graded algebra of modular forms for SL(2,Z) is clearly superior to the lame polynomial algebra
<mlbaker>
because there you really have no idea which basis you 'should' use
<egg|work|egg>
not entirely sure UmbralRaptor would like modular forms as an eggsample, they do require a bit of background before you can have much of an intuition of them
<mlbaker>
i'm just trying to think of something that really forces you to let go of the 'tuples' thing
<mlbaker>
but without being absolutely terrifying like C([0,1]) or something
<egg|work|egg>
for diffgeo you don't want to *completely* let go though, index notation has them hiding in the background
<galois>
oeuf: Your options: zzz, notzzz. My choice: notzzz
<oeuf>
;choose zzz|notzzz
<kmath>
oeuf: notzzz
<oeuf>
hm.
<UmbralRaptor>
Time to deploy 3 additional bots?
<_whitenotifier-3d18>
[Principia] pleroy opened pull request #2141: Synchronize the prognostication when recording and replaying the journal - https://git.io/fj39e
egg|zz|egg has quit [Ping timeout: 183 seconds]
<_whitenotifier-3d18>
[Principia] pleroy edited pull request #2141: Synchronize the prognostication when recording and replaying the journal - https://git.io/fj39e
<_whitenotifier-3d18>
[Principia] pleroy synchronize pull request #2141: Synchronize the prognostication when recording and replaying the journal - https://git.io/fj39e
<mofh>
oeuf: just got out of talks like 15 minutes ago :p
<UmbralRaptor>
SnoopJeDi: did you just have that gif lying around? O_o
<UmbralRaptor>
(also, what movie is that?)
<UmbralRaptor>
I hope that this is just poorly phrased, and means a circular orbit around a black hole with a schwarzschild radius of 2M https://photos.app.goo.gl/SiSZYJTVxbtYbQwUA