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2:48 AM
@Slereah Let $V$ be an $n$-dimensional vector space which is also a $C^\infty$ manifold. For any $v\in V$, we can find a unique natural isomorphism $\phi_v:T_vV\to V$.
Proof. Let $\omega\in V^*$. We can view this as a function $\bar\omega:V\to\Bbb R$ in the sense of manifolds. We seek an isomorphism $\phi_v:T_vV\to V$ such that $\omega(\phi_vw)=\mathrm d\bar\omega(w)$ for each $w\in T_vV$ and $\omega\in V^*$.
For existence, pick a basis $\{\theta^i\}$ of $V^*$ and a basis $\{e_j\}$ of $T_vV$. Then if we define $(\phi_v)_i^j=\mathrm d\bar{\theta^i}(e_j)$ we obtain the desired isomorphism.
For uniqueness, suppose $\psi_v$ has the same properties. Then $\omega((\phi_v-\psi_v)w)=0$ for all $\omega\in V^*,w\in T_vV$. It's obvious that this implies $\phi_v-\psi_v=0$. $\Box$
Now this can be used to turn a Lorentzian vector space $(V,\langle \cdot,\cdot\rangle)$ into a Lorentzian manifold $(V,g)$ if we define $g_v(w,z)=\langle \phi_v w,\phi_v z\rangle$.
This is now one should "properly" define Minkowski space.
@dmckee Thanks for you recent prompt action.
3:35 AM
@yuggib What is Yosida trying to say in the Theorem on page 80? The statement is not what is proved, unless I am misunderstanding what "for any large positive constant $C_1$" is supposed to mean.
Is he trying to say this only works for $C_1$ above some critical value?
Because he seems to be proving that if $|\eta|$ is bounded, so is $|\zeta|$. I checked LH's original paper, his statement is completely different.
1 hour later…
4:39 AM
@JohnRennie Could you solve my question here
Q: What will be pressure of gas measured by manometer if mercury starts to rise on the other side?

Prashant GokhaleWhat will be pressure of gas measured by manometer if mercury starts to rise on the other side? The $P_a$ side is open to the gas container and the $P_0$ side is open to the atmosphere with atmospheric pressure $P_0$. The pressure of gas $P$ here is $P_0 + \rho g H$ where $H$ is as given in i...

@PrashantGokhale Just draw a free body diagram of the mercury column and balance forces.
The total length of mercury in the u-tube will remain constant.
It should'nt be a tough job to write the equations.
5:30 AM
@Sir: Have u received the book yet?
@Secret Hi :-) How dyou post a link and write over it?
The same format you use in the main site when writing link embedded comments
Please note that linebreaks cause the formatting to fail
5:38 AM
When you add a "---" after you've said something, it's an indication that a brief account of "last night's dream" is going to follow :-P
no, I add --- to indicate whatever I said next has nothing to do with the previous message. It forces the chat to break up that chat bubble into 2 so for each bubble I am only focusing one topic at a time
I know. Usually, what u say next is about your dreams is all :-)
I see
Yes, I know this is a Frontier article (and how it is surrounded with controversy including its possible predatory status). But I actually took the time to read this article and the sources in detail, including some breif checks on the background of the authors. So far, everything mentioned there seemed legit, and there are reputable past models of psychology that employed the mathematics of quantum mechanics in the context of psychology
One thing to note that despite they both shared similar mathematical frameworks, there are actually many differences between the phase space of physical systems and psychological, sematic systems:
1. Most sematic, psychological and cognitive systems are highly sensitive to contextual elements. For example, a joke's interpretation is sensitive to the phrasing of the joke and the prior information available to the audience, whereas physical systems, usually their dynamics are controlled by the hamitonian and the hamiltonian contains all information about the evolution of the system
This is why in that article, they mentioned that it is still a work in progress in figuring out how the funniness of jokes interacts with the sematics and phrasing of the joke, (which is why readers will have trouble finding anything that suggests an analogue of schrodinger equation in that article)
@Secret a remarkable find, wild mix of soft/ hard science, thx for sharing. esp intrigued by all the "QM outside of physics" refs 2-28, quite a collection worthy of further perusal.
5:53 AM
actually idk if you guys know about it i just HAD to write about it as it was such a plot twist
write it somewhere that is and this tab was close to me :P
Quantum cognition is an emerging field which applies the mathematical formalism of quantum theory to model cognitive phenomena such as information processing by the human brain, language, decision making, human memory, concepts and conceptual reasoning, human judgment, and perception. The field clearly distinguishes itself from the quantum mind as it is not reliant on the hypothesis that there is something micro-physical quantum mechanical about the brain. Quantum cognition is based on the quantum-like paradigm or generalized quantum paradigm or quantum structure paradigm that informatio...
Information in general for this field of study
Oh, @JohnR: Morning! :-)
Morning :-)
> there are actually many differences between the phase space of physical systems and psychological, sematic systems
Sorry typo, actually there isn't much differences other than the dynamical law cannot be precisely known (yet)
Hmm, I guess we can call a model quantum whenever its mathematical framework contains vector spaces (superposition), noncommutativity (uncertainty principle) and measurement
O btw at the introduction they also mentioned a future plan to develop a QFT inspired model for cognition. However reading this, the first thing that came to my mind is: Well, what will be the analogue of number of particles in a sematic processing of information...?
The dying moments of a red giant
The cool thing about astronomy is it is pretty
6:39 AM
@PrashantGokhale it isn't clear to me what you are asking. The pressure difference is $\Delta P = \rho g h$ as you say in the question, so $P_a = P_0 + \rho g h$. The level of the mercury will start to rise if $P_a$ increases above this value.
6:58 AM
The nature of fluids, even in these seemly mundane scenarios, are extremely important in the context of manufacturing
7:39 AM
Q: How I can see electron

HiggsAnybody can explain me that one can see electron or any other particle by our nicked eyes using any apparatus? Regards

I suppose the OP wants to know why electrons can't be seen with naked eyes.
The initial framing of the question was horrible :-P
It's going to be a part of the HNQ list pretty soon :'D
7:57 AM
@JohnR: Did u take to activities such as these back in Wookey Hole to ease the pain of acute boredom:
(^ Sitting by the railway tracks, waiting for trains to show up to wave to them) Or this perhaps:
@Kaumudi.H wtf is that?
@2017 Seems fine. since from a layman perspective, this is how questions of this type usually implies
7:59 AM
@2017 I've edited my message.
who does that :-P
@Kaumudi.H Ah, I've done that with tyres in my childhood at our ancestral home.
@2017 Bored people of the 50s and 60s :-P
@Kaumudi.H for little children (sub puberty) the countryside around Wookey Hole is a fantastic place to grow up.
Q: Physical explanation of seeing sparkles?

Mihai B.I know this might look more appropriate to ask this kind of question on the biology or medical stackexchange but since this kind of cross-science question I'll ask it here. Sometimes when sitting and working on my computer I get to see sparkles which look like particle trajectories. Sometimes th...

@2017 I've tried but I never got the hang of it :-(
8:00 AM
@JohnRennie :-) I can only imagine.
Well, it is kinda cool to have a cloud chamber in your eyes if that claim really holds
It's hilly and heavily wooded so there are dozens of little (secret!) valleys to play in.
And because it's on the edge of limestone hills there are lots of small caves - big enough to be fun exploring while too small to get lost in.
It's only when I started comparing childhoods with friends who had grown up in cities that I realised just how fantastic a place it is.
It sounds so lovely :-) In my village, the only options are tree-climbing and swimming in the pond.
The problem comes when you get to your mid-teens and want more from life ...
Most of my desires are nonmaterialistic, which is why I tend to be quite content with my life
8:03 AM
Your village? Chennai? :-)
@JohnRennie I understand this; until my teenage years, I used to yearn to go to my village every summer. And then around 14 or 15, I sort of started getting depressed around there.
Chelakkara is a small town in Thrissur district of Kerala. One of the famous pooram ( Vela ) in Trissur dist - in fact in Kerala. Anthimhakalan kavu located in Chelakkara, Which is famous for fire works and traditional celebrations called Kala kali. == Geography == Chelakara is located at 10.70°N 76.35°E / 10.70; 76.35. It has an average elevation of 6 m (20 ft). == Festivals == One of the famous pooram (Vela) in Trissur dist Anthimhakalan kavu located in Chelakkara, which is famous for fire works and traditional celebrations called kala kali. The town has many temples including Narasimhamurthy...
> It has an average elevation of 6 m
Fairly flat (and boring?) landscape then?
There was nothing to do other than tree-climbing and swimming.
8:06 AM
They've literally cut off all the trees now and the pond is beginning to dry up.
I have a sneaking suspicion that 14-18 years olds are always dissatisified. You could put them in a paradise and they'd still be complaining :-) I include the 14-18 year old version of me in this category!
@JohnRennie -__- You oldies, always mocking us teenagers.
That wasn't intended to be mocking. It's just an observation.
Besides, that's not even true.
Not all of us are dissatisfied.
8:09 AM
I'm not even sure it's a bad thing. I think it takes dissatisfied people to change the world. It's probably down to the chronically dissatisfied that we aren't still leaving in caves and hiding from sabre tooth tigers.
Sure sure. I'm merely noting that there is a subtle difference between despairing dissatisfaction and hopeful dissatisfaction.
@JohnR: This is what it looks like:
How to beat a frustrator: Easy, make your choice, but flip the opposite switch, so the frustrator get slapped onto its face by its own rules
Then you win the prediction game, always
8:12 AM
Palm trees :-) I remember trying to climb date palms in Khartoum but the bark is so rough it tears your skin.
Dyou see the yellow building at the back of that picture?
How far away are the hills in the background. Five miles?
@Kaumudi.H Yes?
@JohnRennie That's the hospital in which I was born :-)
Aha - a future famous landmark :-)
@JohnRennie I really dunno. Although they seem close, I doubt if it's only 5 miles...
8:14 AM
I don't know where I was born.
A hospital somewhere in Khartoum I suppose ...
@JohnRennie My mom was born in my house (back in Kerala)
@Kaumudi.H too far to walk then, but within cycling distance?
@JohnRennie I doubt whether it's within cycling distance.
I went cycling one time and never came close.
So you were stuck in the middle of an alluvial plain with no escape possible? :-)
8:18 AM
I usually do enjoy my time over there but three years ago, I had to spend close to 1.5 months over there and that sort of messed me up.
At least one good thing came of it tho--I ended up taking really great pictures before most of the trees were cut down.
Dyou wanna see the pond?
Yes. My house is beside those fields you see. It used to be situated in the midst of a dense forest of rubber trees but they've been cut down now. The pond remains.
8:23 AM
Dyou want to see?
Yes, please post a picture.
It's not so wonderful, I dunno why I built it up but here :-P :
I dunno what these are called but we have these and they're so beautiful:
8:28 AM
(All these pictures I took on a lazy afternoon whilst in the depths of boredom :-P)
I suspect Chennai is less pretty but more fun for someone in their mid to late teens :-)
No, not really; without friends, it gets boring anywhere :-)-:
Yes, I found that as a teen in Wookey Hole. Going to university was my escape route! :-)
Look on the bright side. Life gets better as you get older ... then you die! :-)
1 hour later…
10:11 AM
[Memory correlations] they are a flawed copy of the original ->http://submachine.wikia.com/wiki/Secret
That's a very fun flash game I played in the past
The memory correlations lead to revisiting this game's wiki, and via reading a bunch of articles from layers->karma->karma portal->outer rim -> ... -> eventually result in a new question:
in Mathematics, 21 mins ago, by Secret
What is an example of a space where it is impossible to place any charts and coordinate systems on it?
10:23 AM
anyone here?
what does e + 22 mean?
Figure it out yourself now!
3.086e+22 is a way of writing 3.086 times 10 to the power 22.
or 3.086 + 10 to the power 22 ?
@MartianCactus No, times 10^22
10:29 AM
oh, alright!
The plus is there because it's $3.086 \times 10^{+22}$. If you wrote 3.086E-22 it would mean $3.086 \times 10^{-22}$.
It's a notation that arose in the early days of computers because early displays didn't have a way of writing exponents.
and it is still used in excel
so its just a approximation, right?
10:42 AM
@MartianCactus an approximation? No, it's a way of writing $3.086 \times 10^{+22}$ using ordinary characters that is simple and easy to read.
Hi all
@MartianCactus $3.086 \times 10^{+22}$ has 4 significant digits. So it is indeed an approximate value. You can find a more precise value of megaparsecs (having more significant digits) if you do a bit of googling!
@JohnRennie but a megaparsec isn;t exactly 3.086×10+223.086×10+22, is it?
A megaparsec depends on the sun earth distance
10:46 AM
yeah thats what i am asing
Ah OK. Yes, that figure is an approximation to the number of meters in a megaparsec.
that thing got messed up :P
You will find more precise values over here metric-conversions.org/length/parsecs-to-meters.htm
So it's only as precise as that measurement
oh, thanks!
10:47 AM
Just use the usual formula for propagation of uncertainty
Errors and its propagation is an interesting topic in itself :)
(Most people will disagree about the interesting part :-P)
one thing that bothers me is the notion of significant digit
There's no real physical or mathematical reason for it
It's not even absolute, it depends on the basis you use
how do you calculate
@Slereah It does set a standard for unit conversion, doesn't it ? It is a sort of convention if you say so, just like many other conventions in physics.
yeah but it's often enforced even though it's not actually significant
10:54 AM
$$70,400 m / sec$$ divided by $$3.083 * 10^{22}$$ m?
why does it come in different lines tho?
use {}
and why is one 2 down
for {22}
what i wrote was $$70,400 m / sec$$ divided by3.083 * 10^22 m?
and what can i do to make it come in one line?
11:00 AM
also the unit is $s$ not $sec$
Also start writing it as $ms^{-1}$ rather than $m/s$
It will be very helpful when you do manual calculations
so what is $\frac{70,400m} {sec}$ divided by $3.083 * 10^{22}$ ?
@2017 Also you are supposed to write them upright, not in italic :p
why does it still come on a different line?
is it because im using 2 $ ?
11:06 AM
@Slereah I like the curves ;)
"Unit symbols are printed in roman (upright) type regardless of the type used in the surrounding text. They are printed in lower-case letters unless they are derived from a proper name, in which case the first letter is a capital letter."
Sayeth the official SI standard
oh that works!
Oh wait
"Multiplication must be indicated by a space or a half-high (centred) dot (⋅), since otherwise some prefixes could be misinterpreted as a unit symbol."
$\text{m}\cdot \text{s}^{-1}$
"It is not permissible to use abbreviations for unit symbols or unit names, such as sec (for either s or second), sq. mm (for either mm2 or square millimetre), cc (for either cm3 or cubic centimetre), or mps (for either m/s or metre per second). The use of the correct symbols for SI units, and for units in general, as listed in earlier chapters of this Brochure, is mandatory."
meh :-P
Ban @2017 for crimes against the SI units
11:11 AM
I might still agree to follow that but I am never going to write an upright iota in latex :D
What unit has a iota?
I didn't mean any unit
I meant something like $3+4i$
Imaginary unit isn't part of SI :p
Thank god _/\_
"There are many more non-SI units, which are too numerous to list here, which are
either of historical interest, or are still used but only in specialized fields (for
example, the barrel of oil) or in particular countries (the inch, foot, and yard). The
CIPM can see no case for continuing to use these units in modern scientific and
technical work."
Sorry Americans
You are useless.
11:18 AM
Who sets the SI system rules?
The International System of Units (French: Système international d'unités pronounced: [sistɛm ɛ̃tɛʁnasjɔnal dynite]; abbreviated as SI) is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units. The system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative that began in 1948. It is based on the metre-kilogram-second system...
@2017 The CGPM and the CIPM.
@Loong Thanks!
@2017 the FRENCH
Although unfortunately, the revolutionary calendar never stuck
Mar 12 at 16:05, by Slereah
France was considered by the English to be scandalously loose
If they were loose they should have allowed curves in the units :'D
Oh also you should not use $\text{kB}$
It is $\text{KiB}$
Though technically that isn't part of SI
but it's the CEI norm 60027-2
Wait a moment
11:31 AM
@Slereah When will you change your display picture? :-P
The SI standard says that the dimension of a quantity $Q$ is $\text{dim} Q = \text{L}^\alpha \text{M}^\beta \text{T}^\gamma \text{I}^\delta \Theta^\varepsilon \text{N}^\zeta \text{J}^\eta$
and they say the exponents are integers
But the wavefunction has a non-integer exponent!
@2017 When you will find me a better one
@Slereah I'm too bored. No more SI!!!!
Not actually mine
This one is mine
@Slereah Tell me what type of picture you want!
a Good picture
11:37 AM
Yeah, so this is a good picture ^
11:50 AM
Penrose has arrived!
Damn book is thin as hell
also these guys were so cheap they printed the last page on the cover
and... the first page, too?
Was paying for two extra sheet of papers really that much of a cost
in the above question , I think it should not be 46 it should be 51
Am I correct?
No. 46 is average of 50 and 42.
will it not be average of 42 & 60
Why should it be average of 42 and 60?
as 60 is initial temperature
12:00 PM
This is the average form of NLC
It is cooling from 50 to 42
in the next 10 mins
@2017 yeah
oh I have read it wrong , I thought it is cooling from 60 to 42
12:58 PM
@Slereah Which Penrose?
Techniques of differential topology
It's one of those nasty cheap ass reprints
Never seen this one, looks interesting!
It's kind of a poorly written book, but it contains a lot of proofs you won't find anywhere else
you know, i think it never actually mentions the Einstein equation at any point
The EFE are for plebs stuck in that whole 'problem solving' mentality
yeah I guess it's not that important for causal structures
1:04 PM
His spinor book is a bit crazy
Unless you want to do like energy conditions
@Slereah did you read my proof
Though did you ever used the fact that it was Minkowski space?
Like does it work for any $(\Bbb R^4, g)$
Hmm? I'm just saying you can turn any vector space with a pseudo inner product into a manifold such that each tangent space carries the same product in a natural way.
and vice versa?
1:18 PM
What do you mean by that?
Hm, I'm not sure it makes sense the other way around
eh, whatever
So @0celo7
Is your copy of Penrose also a cheap reprint
2 GSW's!!! WAHOO
What is better, GSW or Polchinski, is the question, I guess GSW
@Slereah I dunno, show me a pic of yours
"By Riemann's theorem"
Jesus Christ can you be a little less specific please
@0celo7 euler's theorem
Ah, the Riemann mapping theorem
1:30 PM
In mathematics and physics, there are a large number of topics named in honor of Swiss mathematician Leonhard Euler (1707-1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. It has been said that...
That's not that bad
I should learn the Riemann mapping theorem
So show me penrose
I don't think mine has shit on the covers though
Mine looks worse :o
What's wrong with yours?
@Slereah I'm 400 miles away
1:34 PM
Well it's alright
but it's fairly cheap
What was it about them saving two pages?
First page is printed right on the cover
same with the last page
I'm guessing a short proof of minkowski being causal is that $x(\lambda)$ has a 0 derivative by the IVT and since it is Minkowski $U^t$ can't be 0 and timelike
1:50 PM
I have a Hilbert space $X$, and $F\subset X$ is not dense. Can I find a $T\in H'$ s.t. $\forall f\in F: T(f)=0$?
@Kaumudi.H 'Fraid not. It's not a business day :/
$F$ is a subspace.
Ah, if $F$ is not dense, then it is strictly contained in a hyperplane.
Let $T$ be the indicator functional for the hyperplane.
Wait, $T$ might not be continuous. Can the orthogonal complement of a closed subspace not be closed?
Orthogonal: definitely not.
It must be closed.
Oh...I'm dumb.
If $F$ is not dense, then there's a vector $g$ in $F^\bot$. Then take $T=$ dual of $g$ by Riesz.
2:14 PM
Q: Did I get hit by a car this morning?

B. BrekkeQuantum mechanics allows us to describe a particle as a wave, and also a collection of particles, which a car happens to be. What separates a typical wave from a classical particle is that the position is not well defined for a wave. I think that it is hard to tell exactly how big an atom is, be...

There is a ridiculously slim but nonzero probability that you get hit
however I don't think I have enough info to write a perfect answer
2:31 PM
@Slereah you could also write down the light cones directly
Then argue it has to cross somewhere
Maybe it doesn't work. But how about showing that the geodesics generating the future light come never enter the past one?
Shouldn't be too difficult since it's all lines
U could use mister Euclid
Hm, I think all minkowski space - a set of finite measure will never be globally hyperbolic
Non finite won't necessarily work tho
Bc $(H, \eta)$
2:50 PM
@2017 I have forgotten the Newton's law of cooling. It says $dQ/dt \propto (T_{sys} - T_{env})$, right? Why's $dQ$ the difference in temperature of water, in there? Just trying to recall how to do these things.
@BalarkaSen dQ=msdT
Ah, right. Thanks a lot.
Then dT is proportional to the RHS..
Probably the thing is like
Minkowski space - the domain of dependance of a Cauchy sirface
Future or past
Is globally hyperbolic
And otherwise not
Wait, what if the spacetime is $(0,\infty)\times\Bbb R$
I think it's still globally hyperbolic
With R the time
Can't think of a simple rule that covers all cases
Might be something like the removed set is convex but only for causal curves
3:10 PM
what are you trying to do?
Trying to think of a rule for minkowski - some points being globally hyperbolic
it's necessary that you can split like $\Bbb R\times\Sigma$
So...you can only remove certain sets
sets of the form $\Bbb R\times A$ I guess
It can be fairly oddly shaped, though
Like a union of light cones
The R x S structure can be non obvious
Ugh, Zorn's lemma has to be wrong
3:27 PM
Well it is based on the axiom of choice
I wonder if there's an equivalent of Zorn's lemma with ~C
Just its negation is fairly not useful
Also I don't know if its negation is true if ~C is true
Doesn't seem to be anything related to partial orders for ~C
3:52 PM
What up peeps, it's yaboy Ethan
Close ya legs, Tan
My management prof gave us an assignment that requires the use of Excel
I told him I don't run Malware on my machine
What did he say?
I use Excel daily
It's wonderful
It's malware
Proprietary software is MALWARE
The only system is GNU and Linux is one of it's many kernels!

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