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5:02 PM
@Secret Second hint: it's trivial
 
@yuggib I thought a geometry proof was trivial
when explaining it to my prof, I figured out it's not
@ACuriousMind I need linear algebra help lol
 
First guess: I only know of two trivial things in mathematics: Identity and zero
Second guess: I don't think it is zero otherwise it sounds too weird
 
I have a product manifold with product metric. I want to show that parallel transport along a curve $P:T_{(p,q)}M\times N\to T_{(r,s)}M\times N$ satisfies $P(T_pM)\subset T_rM$ and $P(T_qN)\subset T_sN$
This should (!) be trivial.
 
@Secret the first one
 
Oh, @ACuriousMind @yuggib
what the fuck is $$\int d\xi\,|\xi\rangle\langle \xi|?$$
 
5:10 PM
@0celo7 the identity (in the form of a sloppy spectral resolution)
 
@yuggib I know its the identiy
but what does it even mean
what is $|\xi\rangle\langle \xi|$ and how do we integrate it
 
a sloppy notation of the spectral theorem
 
what's that?
 
the theorem that characterizes self-adjoint (more generally normal) operators
by means of their spectral properties
 
what does that mean?
@ACuriousMind This theorem is clear when $M$ and $N$ do not have the same dimension if you use some rank-nullity things.
 
5:15 PM
I am not sure what went wrong, because I obviously cannot get the identity here for some reason $\int_{\mathbb{R}}e^{-t^2}\phi(-t)dt=[\sqrt{\pi}\phi(-t)]_{\mathbb{R}} -\int_{\mathbb{R}}\sqrt{\pi}\phi'(-t)dt=\int_{\mathbb{R}}\sqrt{\pi}\phi'(a)da=0$
 
@Secret it is the delta function that behaves as an identity with respect to convolution
 
proof
 
O, I thought you mean that integral act like the identity on the $\phi$
 
@0celo7 go look for it yourself ;-)
 
@yuggib Currently struggling with linear algebraic geometry
will do afterwards
 
5:18 PM
@0celo7 the spectral theorem is one of the basic tools of analysis of operators in hilbert spaces
 
never heard of it.
 
you barely know what a Hilbert space is :-P
 
I obviously think way too hard and missed the point. Ok then, this means my original question should now be solvable time to kill a 1 year old MSE question
 
@yuggib Pretty sure I know?
hmm
it's not clear why $P$ can't evolve an $M$ vector into an $N$ vector at all
 
@0celo7 how many times did you use the characteristic features of a Hilbert space in a proof?
 
5:23 PM
@yuggib 45
 
then you're ignorant if you don't know the spectral theorem
:-P
 
@yuggib it's not mentioned in Jost's analysis
which covers Hilbert spaces
for PDE purposes
 
then Jost's analysis is missing a rather important theorem
even for PDE purposes
I suggest you look it up on either Weidmann's book or, for the basics, in the first volume of Reed and Simon
and I hate my new damn gravatar
 
@yuggib Funny thing is, on my tablet I still see your old one. This gravatar business is very strange
 
indeed
I see my old one as the small icon on my phys.SE pages
but if I go to my profile, the new one is shown
 
5:34 PM
It's also interesting how strongly we can associate random abstract shapes with people. The new ones just don't seem to fit in most cases
 
yes, I am not the deep purple shuriken assembly; I am the light purple fancy fan
 
Lol
 
@ACuriousMind I associate you with Mordin.
 
@0celo7 I associate you with Ed Norton in American History X
 
5:41 PM
@yuggib yikes.
 
@yuggib The fuck?
 
@DanielSank with less swastikas on the body
 
What the hell
 
vzn
@DanielSank buddhism huh? (bernardo has good ideas) :)
 
@vzn Well yes, the Buddhist's had the Nobel Eightfold Path long before physics had the Eightfold Way.
@vzn Well I think you suggested it too.
 
5:45 PM
@0celo7 I'm not meaning this as an offense, I am not meaning that I think you're a nazi
 
You said you associate me with a Nazi
 
vzn
@DanielSank so have you studied buddhism at all?
 
What the hell is that supposed to mean?
 
@vzn Nope.
 
@0celo7 no, I associate you with that type of physical aspect
and facial expressions
 
5:46 PM
So close...
\begin{align}
& \langle(0e^{-y^2}+e^{-x^2})*\delta^2(x,y),\phi\rangle=\underbrace{\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}}_{4}(0e^{-y'^2}+e^{-x'^2})\delta(x)\delta(y)\phi(x-x',y-y')dx'dy'dxdy
\\
& =\underbrace{\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}}_{2}(0e^{-y'^2}+e^{-x'^2})\phi(-x',-y')dx'dy'
\\
& =\int_{\mathbb{R}}e^{-x'^2}\phi(-x',-y')dx'
\\
& =(e^{-(\cdot)^2}*\phi (\cdot,y))(x)
\end{align}

But
$$\mathcal{F}(0e^{-y^2}+e^{-x^2})\cdot \mathcal{F}(\delta^2(x,y))=e^{-x^2}$$

Surely multiply any test function by a gaussian is not the same as convolving a gaussian to the same
 
vzn
@DanielSank try it you might like it, you already know more about it than a lot of physicists :P
 
@0celo7 don't know why
 
vzn
@DanielSank @#$& popsci writing ugh :P
 
@vzn I posted that one because it's well written and delightfully devoid of bs.
 
vzn
@DanielSank lol yeah whatever
 
5:49 PM
@vzn Ok ok, I can see that it looks like I'm strongly biased here. Of course I am.
But seriously, that article is pretty good.
 
@Secret I don't see why you're not satisfied with the result
 
@yuggib ???
 
Look, @vzn, if I were to post links to all of the quantum computing popsci articles I see, you wouldn't have even noticed when I posted this one. I think that proves that I probably didn't post this one just out of self interest.
 
@0celo7 I don't know the true one, so I am making pictorial representations in my mind
 
vzn
@DanielSank the MIT tech review articles are good too dont you think? anyway its all good. think the media is jumping the gun though. "working qm computer by end of 2017"... doubt that martinis has announced that deadline lol
 
5:50 PM
@vzn I am unaware of any such announcement.
 
vzn
@DanielSank people post whatever around here. have posted a lot on qm computing myself over the yrs. posted that article myself, you missed it :P
 
Of course it all depends on what you think "working quantum computer" means.
@vzn I was just arguing that I didn't post this time out of blind self-importance.
 
vzn
whats wrong with self interest anyway? :P
 
^ Nothing
 
if $\phi$ is any test function (satisfying the rapid decay criteria), then multiply it by a gaussian (because a fourier transform of a plain gaussian is just a gaussian) will not be the same as convolving said function with a gaussian (which from the integration by parts did above, is 0)?

convolution theorem says $\mathcal{F}(f*g)=\mathcal{F}(f)\cdot\mathcal{F}(g)$
 
vzn
5:51 PM
so then you have the humility associated with/ required for buddhism :P
 
@ACuriousMind is sort of like BG's Kagain but with a red beard
@Secret if you want to take the Fourier transform, you have to take the Fourier transform of everything, test function included
 
@yuggib ACuriousMind is like one of the BeeGees???
 
and the Fourier transform of a test function is in general different from the original test function
 
o crap, that changes everything. So it seems my 3rd year optics professor is massively and sloppily simplified the fourier pairs and convolution theorem

Thanks yuggib, both me and my professor have been stumped by this question for basically a year. now its clear
 
@JohnRennie no, like a Baldur's Gate character
@Secret no prob
 
5:55 PM
@yuggib You're insane
 
will write up the answer to the question tmr
 
@0celo7 maybe
or I'm on drugs like you yesterday ;-)
 
Hello
Can I ask a question?
 
Fun fact: The phenomenon "stumped" can be contagious. For example, if you have an ultra hard qustion and you ask your professor, if he/she too does not knwo the answer, then you have two stumped people
@Mattew shoot
as in, shoot and ask the question (if you are not familar with the slang)
 
5:59 PM
Ah ok
 
testing
testing completed
 
I'm 14 and I'm in an high school where I study languages, I study math and I like it, btw I'm not very good at it. this years I'll study physics, Do I need to be very good in math to study physics?
 
Jim
0
Q: Can a balloon chain to space?

TrollThe longest balloon chain is recorded at 20 km. I want to know has any one tried this with blimps and rope? Can a balloon chain be up scaled to reach space with enough tension to carry a payload? Wind is null at equator. Weight of rope is divided by blimps. Gravity and centrifugal force of the Ea...

what happened to the original of this?
also, this isn't acceptable practice
 
@Mattew You need to be quite comfortable with calculus for high school physics. As you continue, you will learn more sophisticated maths. While you don't necessary need to know how to formulate a proof, you need to understand how to apply the mathematics in computing the physics and understand the models
 
@Jim As I said yesterday:
18 hours ago, by ACuriousMind
In unrelated news, I see this user finally identifies as what they are :P
 
Jim
6:12 PM
@ACuriousMind wise, wise words
 
@yuggib lol
@JohnRennie lol, again
@Mattew You do need math to do physics, but you shouldn't worry if you're not the best at it. You don't need to excel at it, you just need to be reasonably competent with it.
@DanielSank I do hope you will fill it with content - I tried starting a blog once or twice and always ran out of steam quickly
 
Last night dream is long. However the part that is relevant to h bar is the following weird object:

Let A = a 5x5 square matrix

$\vec{T}=\sum_{i=1}^5(\hat{e_i}A\hat{e_i})\hat{e_i}$

a vector of diagonal elements
O, and in the dream, 0celo7 and yuggib mentioned the following things as we started the discussion
 
when the fuck was that
@ACuriousMind Ok, please humor me, I'm going insane.
We have the product Riemannian manifold $M\times N$
 
DW 2/9/2016 ~3:00, i.e. photoshopped just to illustrate what happened in last night dream
 
Suppose I take a vector $v=(x,0)$
i.e. it only lies in the $M$ tangent space
@ACuriousMind does that make sense?
@Secret Please don't photoshop what I say.
 
6:27 PM
@0celo7 Yeah
 
@ACuriousMind can parallel transport rotate $v$ into the $N$ tangent space at some other point?
 
I feel that it should not
 
ok, will mention similar things in plain text in the future then.
 
For the life of me I cannot prove that.
not at the same point, but let's consider that
suppose we parallel transport along a loop
 
hi folks, any one wish to provide answer on WB about solar sails worldbuilding.stackexchange.com/q/52675/20315 , solving few equations for 500 WB rep ?
 
6:28 PM
The connection forms (Christoffels) are the direct sum of the individual ones, aren't the`y?
 
@ACuriousMind In a certain sense, yes.
 
what you think about
 
@ACuriousMind the curve we are going along is nontrivial in $M$ and $N$
i.e. its of the form $(\gamma(t),\lambda(t))$ where both curves are nontrivial.
 
After my post in that mock screenshot, we then have a couple more messages (details forgot), and then I posted an animated GIF of that stick diagram, where each spoke is supposed to represent the magnitude of each diagonal element. The GIF have the spokes extend/contract in some uniform manner
 
but let's think about loops for a moment
 
6:31 PM
When I woke up, the first thing that came to mind is the invariance of the trace under a change of basis
 
user54412
@JohnRennie Back in college I had a number of informal chats with one of the founders of QCD, and he was of the same opinion. It seems a lot of quantum people start thinking like that, but nothing ever comes of alternative formulations.
 
As for what Dream 0celo7 mentioned about norrin, he said that they are matrices with positive trace being used as a metric on a topology, that is the values of the distance function $d(x,y)$ is the trace of some matrix assigned uniquely to each pair of points in the topology
 
@ACuriousMind The parallel transport eq. is linear
but even if the initial data lie only along $M$, there is no reason why the solution can't be nonzero on $N$
it could evolve to be nonzero there
 
You know, for the holonomy of a principal bundle, I would write the holonomy as $\mathrm{e}^{\int_\gamma A_1 + A_2} = \mathrm{e}^{\int_\gamma A_1}\mathrm{e}^{\int_\gamma A_2}$ and we would be done (since the exponential of a matrix leaves the same subspaces invariant as the original matrix), modulo you demanding rigorous proof that that's the holonomy :P
 
fucking matrix exponential.
RIEMANNIAN GEOMETRY
 
6:36 PM
I have no idea how to prove or even think about this with "Riemannian" methods
 
My prof said the same thing
:(
brb class
 
Idea extraction: I have no idea why would anyone want to assign a positive trace matrix to every pair of points in a topology as a way to define a metric, nor on the top of my head whether this will be of any use
 
You want to topologize the space of matrices, clearly.
 
My dreams have been doing a lot of maths that are logically consistent, but makes no sense lately
 
user54412
I'm voting to close this question as off-topic because should be for aviation not physics. — Troll 17 mins ago
 
Jim
6:41 PM
@ChrisWhite saw that. Seems like someone is either bitter or looking for a reason to complain
or both
 
user54412
I don't suppose the consistent history of terrible posts, combined with the user name, is enough to warrant nuking the account for good?
 
Jim
not being a mod, I have no idea
 
I think that user was already once banned for "low-quality contributions"; no reason we can't do it again
 
Jim
But I suspect the user's name is never something taken into consideration
except to see if they're a repeat offender
 
@Jim They don't need the name for that, users are mod-only-visibly "annotated" with their suspension history and other notices
 
Jim
6:51 PM
then yeah names aren't taken into it.
Setting your user name to "Troll" doesn't make you a troll any more than setting your user name to "Hitler" makes you Hitler
Although, setting your user name to "Jim" does make you my slave/subject
 
user54412
I would argue both make you a troll
 
@Jim Still, both make me negatively predisposed toward the user with that name.
 
@ACuriousMind Why do you hate Jim
 
Jim
@ACuriousMind "Einstein"!=Einstein?
 
@0celo7 what?
I love Jim
 
Jim
6:54 PM
<3
 
Daily reminder Einstein was a crackpot.
 
So are you.
 
@BalarkaSen He can't be a crackpot, he's not doing any physics.
 
@BalarkaSen My holonomy proof is wrong
 
He's a crackettle. Kettle calling pot this time.
 
6:55 PM
I do smoke a lot of crack
 
Jim
@0celo7 You could be my town's new mayor!
 
@Jim I would explode if I entered Canada
 
Jim
Only one way to find out
 
I know for certain
I cannot leave the USA
 
@0celo7 What part of you did you push past the Canadian border to find out? :P
 
Jim
6:57 PM
Not with that attitude
 
@ACuriousMind I used to be a Siamese twin.
 
Jim
@0celo7 Are you sure that wasn't just the crack?
 
^good point
 
No, my crack habit started later
 
6:59 PM
Stop smoking. Start injecting stuff into your veins.
 
Jim
out of grief. I understand
 
@BalarkaSen I inject chern's blood
Cloned blood
 
Jim
@0celo7 Our old crack-smoking mayor wasn't around much either. You could be the new crack-smoking mayor from a distance
 
@Jim I hate Canada
I don't like hats
And I hate socialist hats
QED I hate Canada
 
Jim
Okay, we're sorry, eh?
 
user54412
7:02 PM
@ACuriousMind so my friends have pointed out that the new PoE league starts in an hour
 
PoE?
 
@ChrisWhite I know! Mainly improvements of the endgame content when I already was too bad to beat high tier maps ;/
 
Jim
Not sure what makes us particularly socialist. But our culture is primarily sports that involve a lot of full-body contact, drinking beer, and talking about the weather. You don't like those things?
 
Sports are for hooligans
Beer is for alcoholics
Weather is for people who go outside
So no, I don't like them.
 
And full-body contact is for...?
 
7:05 PM
Reproduction
 
Jim
touche
we also have the worst traffic anywhere. You like being stuck in traffic?
 
user54412
@Jim California challenges you
 
Implying Canada traffic is worse than DC, Atlanta or LA
 
Jim
The part of Highway 401 that passes through Toronto is the world's busiest highway,[4][5] and one of the widest.[6][7]
^ from wikipedia
 
Busiest does not mean worst traffic @Jim
 
7:07 PM
Widest doesn't go well with traffic jam.
 
Jim
@0celo7 in rush hours it does
 
I bet Beijing has worse traffic
New Dheli
 
@0celo7 It's difficult to tell in all that smog :P
 
user54412
There was a maneuver near LA I did regularly that involved 11 lane shifts in the space of a mile.
 
I have heard Dehli's got some bad traffic, yes.
 
7:10 PM
@ChrisWhite I crossed 7 lanes in one shot in Atlanta. It was thrilling.
Almost died
@ACuriousMind I found it.
 
Jim
@ChrisWhite can't do that many lane shifts on the 401 in a mile. I drive it every day. Making 5 lane shifts in 2 km without getting into an accident is nearly impossible. Everyday, some new fool will try though
 
People with baby faces who have mustaches.
My rage cannot be contained in this small chat box.
You could be a skinny short emo with a beanie in Baghdad
But the guy with the mustache makes me infinitely more angry
 
You have issues
 
...am I a misanthrope
 
@0celo7 No, just suprisingly superficial in your judgement of people :P
 
7:14 PM
Surprisingly?
 
Okay, maybe not that surprisingly :P
 
Wooooow
This holonomy problem is going to kill me
I have no intuition for product metrics.
@ACuriousMind I have good reasons for my judgements
@ACuriousMind Sandals with long pants.
 
@ACuriousMind Well, the mission statement includes not using it for in-depth technical writing. I think I'd rather use it as a gateway to the (huge amount) of stuff I've already written.
My github repo is full of tons of stuff I need to get exposed.
 
Jim
@BalarkaSen I looked into that further. It's called wide because some sections have 18 lanes. But that's misleading because it's really only 8 lanes in the express section with 10 in the collector section that feeds directly into the express every couple km. So 500,000 cars in a day are crowded into the 8 lane sections for the majority of the trip. Add to the the carpool lane and, if you're a single person in a car, you're stuck to 6 lanes. That's how you get traffic jams at 1pm on a Wednesday
 
18 lanes
that's quite a lot
 
7:29 PM
Whoa. You even researched on that.
 
Jim
including both directions though
and that's only in the city. Carnage alley has only 3 lanes per direction
 
@Jim 500,000 cars on 8 lanes is not good.
 
Why not?
 
That's way too much traffic.
 
Why?
 
Jim
7:32 PM
have you ever said "I'm going to leave work early to beat the traffic"?
Nobody that takes the 401 thinks that. Traffic doesn't end between 6am and 10pm
 
No, I walk home from work.
 
Jim
@0celo7 might be faster if I did that
 
Ok, maybe I should prove some easier facts about holonomy.
Different points have isomorphic holonomies.
connect the two points with a curve, parallel transport along it
sure
@ACuriousMind Hmm.
@ACuriousMind When can I block diagonalize a matrix?
more than one block obviously :)
 
@0celo7 If the space decomposes into the direct sum of invariant subspaces.
 
@ACuriousMind I knew that. Nothing better?
I have Latin and Greek letters
what's a good third alphabet
@BalarkaSen please teach me Hindi letters
I need more indices
 
Jim
7:42 PM
Russian?
 
@0celo7 Not that I know
@0celo7 Hebrew
 
Jim
^ that
 
@ACuriousMind teach me your Jewish ways pls
I've made a curious discovery.
 
Pretty sure LaTeX has a package for its use in math mode, where else do the logicians get their alephs from?
 
But I need more letters to prove it!
isn't aleph built in
 
7:44 PM
damn, it is
 
@ACuriousMind n00b
 
But the beth isn't, and I've seen that, too
 
I should buy Besse Einstein manifolds
Geometry is just too addictive
and geometry books are even more
 
Jim
@0celo7 and therein lies the most fundamental difference between you and I
 
Yes?
 
Jim
7:47 PM
I don't find it that interesting
 
Have you done any geometry that wasn't physics
 
Jim
None that you'd count as actually doing geometry
 
There's your problem
 
Jim
I think of it as a solution
 
@ACuriousMind I NEED MORE LETTERS
ahhhh
I want to type my thought but I cannot
 
Jim
7:49 PM
side note: I learned something new today
0
A: how dense fluid affect the buoyancy force?

Dr. SmithDogs are able to float on clouds with buoyancy of fluid, because their so light

 
@ACuriousMind Screw it, let's go string theory
 
...who voted that up?
Also, lol
 
My convention is $M=(m,\mu)$, does that make sense?
 
Jim
@ACuriousMind no idea
 
big Latin for $M\times N$, small Latin for $M$, Greek for $N$
@ACuriousMind The parallel transport equation for $M\times N$ is $\dot v^I+\Gamma^I_{JK}v^J\dot c^K=0$ along $c(t)$
Now, suppose I pick $I=i$, i.e. I'm looking at the component along the first manifold
Then I think it collapses to $\dot v^i+\Gamma^i_{jk}v^j\dot c^k=0$
and now we only take information from $M$
I'll have to check that to be sure
IN PARTICULAR
suppose we go along the loop $(\gamma(t),q)$
then we should definitely stay in the $M$ tangent spaces
but this loop is the same parallel transport as $(\gamma(t),\lambda(t))$
(on vectors in $T_pM$)
So a general curve has to leave $T_pM$ invariant!
Because that special one definitely does
And they should all be equal
How insane/plausible am I sounding?
 
Jim
7:54 PM
very
 
@Jim do you have the formula for Chistoffels memorized
 
Jim
@0celo7 good lord, no. Why would I? I mean, I know the basic form of it off the top of my head, but why would I memorize it? I only have to use it maybe once or twice a year.
 
Huh, I use it more frequently than that.
Maybe you're not doing real GR.
@ACuriousMind Please please please is $(A\oplus B)^{-1}=A^{-1}\oplus B^{-1}$
 
You can work that out yourself.
 
@ACuriousMind How? I don't know how to invert matrices.
 
8:00 PM
That's your problem, not mine
 
wtf
 
Jim
@0celo7 I do a lot of numerical relativity. I use it only once or twice because I program it into a new code that often and let the computer calculate it when needed.
 
I think you need it if you want to check the splitting of holonomy groups.
 
😂😂 Some questions...
 
8:07 PM
NO EMOJIS
 
Jim
-2
A: Does cosmic activity outside our observable universe affect us?

Mihai BarboiIf you consider entanglement and or wormholes, there is no such thing as completely isolated object (from Multiverse perspective which some theorists consider to be the result of String theory ) They all depend on each others. Nothing is completely isolated from the rest. The isolation is only b...

quantum people, have at him
 
@ACuriousMind Eh, I have a very unconvincing proof now
I will type it up
 
@Jim Are quantum people in a superposition of being people and being not-people?
 
Jim
@ACuriousMind also, yes
 
@ACuriousMind Erm, how do I make a newcommand in TeX have an input?
I want to write \Hol{p} for \mathrm{Hol}_p.
 
8:13 PM
@0celo7 Did you try googling "newcommand tex argument"?
 
...didn't think about it, no
 
!!?!!!?!???!?!?!??!!
 
what?
I googled and it's not helping ;_;
\newcommand{\Hol}[1]{\operatorname{Hol}_{#1}}
Aha!
 
Apparently, it helped :P
 
This proof is so nasty, assuming it works.
 
8:17 PM
why Entanglement cannot transfer information? Don't I know the result of one photon by another photon in the quantum eraser experiment?
 
coordinates, ugh
 
11
Q: Why can't I use Bell's Theorem for faster than light communication?

ikeI read this description of Bell's theorem. I understand he's restating it slightly, so there may be incorrect assumptions there, or I may have some. I think Bell's theorem should lead to FTL communication, and I'll try to lay out my assumptions as to how, and propose an experiment. So: A and B ...

Search for "no-communication theorem" on the site or on Google for further explanation
 
@ACuriousMind thanks, I will look it up
 
Jim
Just read the article. It's interesting.
Not definitive, but interesting. Much like the superluminal neutrinos, but it's from 2006. If this were true, I'd say we'd all know about it by now
 
@Jim not really
@ACuriousMind what ever happened to that paper that claimed to have a nonperturbative formulation of gravity in QFT that was renormalizable?
or something like that
it was mentioned here a few times.
 
8:30 PM
@0celo7 That was a series of papers by Hennaux and others that proved non-perturbative renormalizability of certain gauge theories, and causally mentioned that GR-as-a-gauge-theory falls into that class of theories. I haven't been able to determine the significance of these results, but thanks for reminding me that I wanted to ask a question about that.
 
@ACuriousMind yw
 
?
Sigh...I first have to clean up my download folder to find those papers again, though :/
 
@ACuriousMind y = you're, w = welcome
 
in return you can read my ramblings on holonomy
no balls I promise
(I take that back, I have to trivialize a bundle and you can always do that over a geodesic ball >:3)
 
8:47 PM
@ACuriousMind Aha!
I'm pretty sure it does preserve subspaces.
I'm using rank-nullity, you'd be proud
I take it back
ugh
@ACuriousMind Suppose I have a linear map $P:A\oplus B\to C\oplus D$
Suppose that $P$ maps $A\oplus 0$ isomorphically onto $C\oplus 0$
--- to be continued later.
@ACuriousMind Is "parallely" a word?
 

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