The h Bar

Secret

13:08

first time heard of such term

physicists also used it in particle physics according to some arxivs

This one single book chapter combines a lot of chemistry physics and maths

I wonder if you guys have seen something similar in the quantum texts (except for virial and force of course)

The continuity theorem is interesting, since the commutator term vanishes. Granted, within an atom, we won't expect the number density to change

bolbteppa

13:25

Looks like engineery/chemistry-ey versions of physics things

Secret

indeed, Bader is putting the chemistry in a (mostly) physics language

bolbteppa

'the addition theorem: $\oint (\mathbf{\mathcal{P}}^{A^{A}} + \mathbf{\mathcal{Q}}^{B^{B}}) \cdot d \mathbf{ \mathcal{S}} = \oint \mathbf{\mathcal{P}}^{A^{A}} \cdot d \mathbf{ \mathcal{S}} + \oint \mathbf{\mathcal{Q}}^{B^{B}} \cdot d \mathbf{ \mathcal{S}}$'

Secret

uh, that's a trivial property of $\oint$...

bolbteppa

zactly

Secret

Anyway, his theory had been widely used among the literature since the conception and is used to predict reaction pathways, and bonding in molecules e.g.:

pubs.acs.org/doi/10.1021/acs.organomet.7b00234

Secret

15:46

1

Q: Relation between Heisenberg uncertainty principle and relativity

In special relativity light changes time, mass, space for maintaining it constant speed.Heisenberg uncertainty principle is for everything even light. If light change those things for maintaining it's own constant speed, means it can make momentum and positions of particles uncertain for maintain...

This is a strange question, why would two conjugated observables must imply there is an invariant governing their tradeoffs?

Sir Cumference

scienceblogs.com/evolutionblog/2008/07/28/…

Found this fascinating

> ...there is a distinction between understanding conceptually what mathematics is all about on the one hand, and being able to carry out the mechanics of solving actual problems on the other. When teachers say, “Multiplication is repeated addition,” or “Exponentiation is repeated multiplication,” they are addressing the mechanical aspects of the subject.

> [The concern is that] these mechanical considerations sometimes transgress their boundaries, and get improperly applied to the conceptual side of things. This makes it difficult to give students a broader understanding of what these operations are really all about.

Slereah

You can't do perturbation theory around the free theory, but can you perturb it around another value?

Like if you have some coupling constant $g$, can you do perturbation theory around $g' \neq 0$

Anonymous

@SirCumference We already discussed about this before I guess. When teachers talk about multiplication as "repeated addition" they are actually talking about multiplication for the whole numbers only. There's a systematic way to define it for integers, rationals, reals and so on...

Anonymous

But yeah, I agree with you that they should mention it explicitly

Sir Cumference

@Blue Yeah, I just think it's interesting to separate the mechanics of mathematics from the concepts

You can never really be satisfied with definitions in mathematics until you take much higher level classes

Anonymous

16:03

That is very true. For example I'm almost fed up with my math professor teaching us tons of artificial methods to solve DEs without inducing motivation (typical for an engineering course). That is perhaps why so many people grow up to hate math.

Sir Cumference

@Blue Exactly. If we're to speak about the philosophy of mathematics, you won't get a good explanation on what something as (seemingly) simple as multiplication is unless you can understand

> Abstractly speaking, multiplication is one of the basic operations that define the algebraic structure known as a “ring.” Rings are required to have two binary operations. One of these operations is called addition, and it is required to satisfy certain axioms. Multiplication is a second operation that is required only to be associative, and to interact with addition via the distributive property.

I remember when I first took calculus, I had no ability to grasp what a differential was. The only response I got was "they are properly defined in differential geometry"

bolbteppa

How To ABSORB TEXTBOOKS Like A Sponge

►

Anonymous

@SirCumference I'm not sure that's a very good definition for multiplication of numbers considering that multiplication elements of a ring may not be commutative. :p It probably needs some fixing

bolbteppa

I never read the first and last sentence of every paragraph to help build a map, that might help

Anonymous

@SirCumference Heh. I was just told it is a "small change"

Anonymous

16:10

in $x,y,..$

Balarka Sen

Understanding the rigorous definition of a differential does indeed require some proper background in multivariable analysis.

A "small change" is not that bad.

Anonymous

@BalarkaSen I never said it is bad. But I find it nice that SirC's teacher mentioned that explicitly.

Balarka Sen

Mathematics is a subtle balance of rigor and intuition. It's important to understand what the proportion of either is

Anonymous

I agree. Maybe I'm a bit biased because two of physicists I really look upto, always state clearly when they are shoving things under the rug in order to focus more on the intuitive aspects of the topic. That's the the type of teaching I'd love people to emulate (so that students don't develop misconceptions after dealing with flawed analogies over extended periods of time).

Anonymous

Probably I'll never be half as good a physicist as they are. But that's a quality I really appreciate.

Balarka Sen

16:29

in an ideal world mathematical communications would not be limited to writing "Theorem 4.2: Blah" and "Proof: Da da da \blacksquare"

because that's most inefficient at communicating the thought that goes into writing a proof

277

Q: Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, or describe how they are connected for you. I've been fascinated by the phenomenon the que...

soft-question ho.history-overview big-list big-picture math-communication

Slereah

It's the Stone-Von Neumann theorem that guarantees that all QM theories are on the same Hilbert space, right?

Hence perturbation theory

Abcd

$PV = \mathbb{constant}$ only for reversible isothermal process? Or is it valid for both reversible and irreversible isothermal processes?

John Rennie

@Abcd assuming we are talking about an ideal gas then at equilibrium it is always true that $PV = nRT$, so for an isothermal process where $T$ is a constant $PV$ will also be constant.

But ...

Irreversible processes by definition cause the system to be out of equilibrium

Abcd

@JohnRennie oh, I didn't know that condition.

John Rennie

Now we have a problem because if the system is not at equilbrium properties like the pressure and temperature are strictly speaking not defined.

If the temperature at equilibrium before the change was $T$, and the temperature at equilibrium after the change has the same value of $T$ then we can be sure that $PV$ is the same before and after the change.

But if the change goes through highly non-equilbrium configurations then we can't say much about what value $PV$ has during the change.

Abcd

16:45

Okay.

John Rennie

Although in most cases the system doesn't get far from equilibrium and $PV$ is approximately constant.

CooperCape

17:18

The disappointment I have in my brother is immeasurable when I see a facebook post from him saying "Logan Paul just posted."

Balarka Sen

tfw your brother is a Maverick

Anonymous

@CooperCape Does the sentence end at just that?

CooperCape

Yes, yes it does.

Balarka Sen

absolute savage

CooperCape

won't lie he does have maverick merch

Balarka Sen

17:21

loool

John Rennie

My niece seems completely uninterested in Facebook. Apparently she prefers real life (whatever that is :-)

Anonymous

@CooperCape Well, good to know. I'm watching that video currently. :p

skullpatrol

Good for her :-)

CooperCape

I only use facebook for the messenger tbh

Anonymous

Don't know if it's just a pretence, but it seems to be unlike his other videos.

CooperCape

17:24

huh damn

Anonymous

@CooperCape Same here. That too, just for conversing with only one or two friends

John Rennie

@skullpatrol one day she'll discover that real life is just an illusion caused by insufficient drugs and alcohol.

skullpatrol

:-)

CooperCape

classic John on the xannies again

Balarka Sen

his recent video is ok

CooperCape

17:25

Yeah I really only message one person on the reg

Did you see the one with the guy who committed suicide?

Balarka Sen

who hasn't?

Anonymous

Doesn't John's niece want to be a journalist? I suspect that's why she prefers real life :D

Balarka Sen

I prefer real death

CooperCape

Classy

Anonymous

I prefer any life where I don't have to face frequent human contact.

2

CooperCape

17:28

keeps life simple

skullpatrol

yeah, simply boring

John Rennie

@Blue Hell is other people

... and hydrodynamics. Hell is hydrodynamics too.

CooperCape

ewww skull likes human contact...

skullpatrol

isn't what we're doing here "human contact?"

John Rennie

@skullpatrol what exactly are you doing with your PC?

Balarka Sen

17:31

I only like human symplectic, not human contact

CooperCape

nah I'm indoors and alone it's chill

skullpatrol

there are sites you can chat with a bot

Balarka Sen

@skull That's what we are doing, with you, right now

vzn

← contrarian view, hydrodynamics is utopia

Balarka Sen

Ohhh shite, the arrow again

← vague implications intensifies

skullpatrol

17:33

<----- :O

vzn

↑ anti arrows :(

skullpatrol

don't get the wrong idea, we appreciate your arrow notation

vzn

appears BaSe is an arrow hater :(

skullpatrol

he's just a hater in general

Balarka Sen

← Dabs in arrows

vzn

17:36

@skullpatrol lol agreed o_O ...(maybe needs hot gf to regain lust for life?)

Balarka Sen

←O←

Dab

@vzn This is discrimination. I do not identify as a heterosexual male.

skullpatrol

he hates life

11 mins ago, by Balarka Sen

I prefer real death

vzn

so BaSe are you communist or what?

Balarka Sen

Far communist

vzn

@BalarkaSen do you identify as something in particular? (prepares for ridiculous response)

Balarka Sen

17:39

I'm transfluid non-binary pansexual.avi

►

Anonymous

@vzn This ^ (Speaking on BS's behalf)

vzn

@BalarkaSen miley cyrus & kristen stewart also identify as fluid and/ or pansexual so youre in good company :P

skullpatrol

That's me on the right.

With my cheap sunglasses, oh ya...

vzn

@BalarkaSen did you cry real tears when vine died? :P

Balarka Sen

17:42

yes, i am hyped for vine 2

more 3 second videos of lele pons smashing into concrete

thats what i call content

skullpatrol

this is what I call content

Balarka Sen

@vzn The thing I posted is not a vine, actually. I do not know if it was made in vine or in youtube, but it's origin lies in the infamous Idubbbz video where he jumps off the kitchen chimney in his leather green suit and cries out "I'm gay" with shrieks of Filthy Frank from behind the scene

vzn

@Blue alas not cool/ hip enough to recognize it, so now desperately looking for algorithmic help eg recode.net/2015/12/23/11621694/… inc.com/minda-zetlin/…

Balarka Sen

good days

vzn

@BalarkaSen who is he? think nobody can compare to jake paul & his brother these days :P

Balarka Sen

17:47

Idubbbz? An edgy youtube comedian

Anonymous

@vzn Well, I got it from here: massappeal.com/psychedelic-documentary-soviet-hippies :-d

vzn

@skullpatrol metrolyrics.com/cheap-sunglasses-lyrics-zz-top.html

bolbteppa

I have fluctuated between thinking of LSZ is the most, then least, and now again most, important concept in QFT

vzn

@Blue yikes so maybe you are communist sympathizer also? o_O

Balarka Sen

Soviet Russia did indeed have a serious hipster culture in the later days

skullpatrol

17:50

5 mins ago, by skullpatrol

this is what I call content

bolbteppa

The whole hand-waving about what goes on in between the start and end being some magic is pretty much an intrinsic part of QFT

What does Weinberg say on LSZ Vs. Srednicki, seems like he's going overboard

Balarka Sen

@Blue Do you know/want to learn the singular value decomposition? I never learnt it but I may have to

Anonymous

@vzn Not really

vzn

@skullpatrol so zz top fan? have you heard their new one "27 lighters"? o_O ... like the video, play it in new guitar hero

Anonymous

@BalarkaSen It's given in Axler. I read it briefly but forgot it. Have to re-read Axler again

skullpatrol

17:52

@vzn the older stuff

Balarka Sen

I think it's in Ted's book

Let's see

bolbteppa

Axler probably has a 10 minute video on it

Balarka Sen

Hm no

bolbteppa

I have just skimmed what he says about it and want to read it

Anonymous

I'm focusing on the abstract algebra part more now. But sure, we can do it sometime this week

bolbteppa

17:53

Singular Value Decomposition

►

Balarka Sen

@Blue I think I'm going to learn it right now. It shouldn't take long

Thanks @bolbteppa

Anonymous

@BalarkaSen Okay, I'm opening the book

Anonymous

Let's see

Anonymous

Reading...

vzn

@skullpatrol are you psyched about superbowl?

skullpatrol

17:55

Yup, the big question is will they cover.

–7 points

Balarka Sen

Eigenvalues of $A^T A$, is it? Wonder what that means

vzn

so BaSe wanted to ask, were you serious about this? always hard to tell with you

Jan 20 at 19:18, by Balarka Sen

@vzn My close family members have schizophrenia (and it's provably inheritable, so I have a sufficiently nontrivial probability of having schizophrenia later in life). Not even joking.

Balarka Sen

Yes, that's true @vzn

Anonymous

Okay so if $T$ is a linear map on $V$ (an endomorphism)....

vzn

@BalarkaSen not brothers or sisters? have studied sch. myself some, very complex

Anonymous

17:56

The singular values of $T$ are the eigenvalues of $\sqrt{T * T}$

bolbteppa

I am shocked I never realized if you treat the entries of a matrix as the value of a bunch of polynomials in the entries, and then ignore the value and expand the matrix in terms of polynomials and end up with matrix polynomials, you can completely ignore vector space theory and get big results on matrices a completely different and easier way

Anonymous

With each eigenvalue listed the same number of times as the dimension of the null space of $\sqrt{T * T} - \lambda I$

Anonymous

Interesting

Anonymous

Proof now

Balarka Sen

I do not understand the meaning of this

skullpatrol

17:59

@vzn the jaguars almost beat NE and the Eagles have the momentum going into the game, the 7 point spread will make it....interesting :-)

vzn

@skullpatrol only interested in the commercials + Timberlake :P j/k

skullpatrol

:-D

Balarka Sen

If $A$ is orthogonal, $A^\mathsf{T} A = I$

skullpatrol

Anyway pro-bowl today

Anonymous

@BalarkaSen $T^{*}$ is the adjoint of $T$

vzn

18:01

@skullpatrol so youre a mystery, did you ever study physics in school?

Anonymous

Is that the confusion?

Balarka Sen

I have no confusion. I am just saying, what does this notion mean?

The first question to ask when you're learning any piece of mathematics, as usual

So anyway, orthogonal matrices have singular value 1

What happens when you're not orthogonal?

Googling around tells me singular values are the lengths of the semi-major axes of the ellipse $A S$ where $S$ is the unit sphere centered at the origin.

That's strange, and interesting

dmckee

I can't be the only one who thinks that "What is [quantum thingy], really?" translates into English as "Tell me I don't have to deal with the ambiguity of quantum and mathematical complexity of quantum mechanics". Can, I?

3

Anonymous

24

A: How can you explain the Singular Value Decomposition to Non-specialists?

Much of linear algebra is about linear operators, that is, linear transformations of one space to itself. A typical result is that by choosing a suitable basis for the space, the operator can be expressed in a simple matrix form, for instance, diagonal. However, this does not apply to all operato...

Anonymous

This looks good

Balarka Sen

18:12

I see what's happening. Choose a orthonormal basis $\{e_1, \cdots, e_n\}$ of $\Bbb R^n$

Preferably the standard basis

Anonymous

The singular value decomposition is the only main result about linear transformations between two different spaces. It says that by choosing suitable bases for the spaces, the transformation can be expressed in a simple matrix form, a diagonal matrix. And this works for all linear transformations. Moreover, the bases are very nice: orthogonal bases.

Balarka Sen

This gets send to the basis $\{Ae_1, \cdots, Ae_n\}$

The "envelop" is the ellipsoid that $A$ sends the unit sphere to

Well, suppose $A^T A$ is diagonal

Then $A^T A e_i = \lambda_i e_i$

So $A^T$ sends our new basis $\{Ae_1, \cdots, Ae_n\}$ to $\{\lambda_1 e_1, \cdots, e_n \lambda_n\}$

Anonymous

demonstrations.wolfram.com/SingularValueDecomposition

Anonymous

@BalarkaSen Ah, yes. In 2D we can visualize it

Balarka Sen

there is nothing special about 2 dimensions

Anonymous

18:17

Just consider $\hat{i},\hat{j}$

Balarka Sen

Oh, I get it, I get it. If $A$ stretches the $i$-th semi-axis of the ellipse by a factor of $\sigma_i$, then $A^\mathsf{T} A$ stretches it by $\sigma_i^2$

So $\lambda_i = \sigma_i^2$

This $\sigma_i$ is the singular value of $A$. It's the square root of the eigenvalue $\lambda_i$ of $A^T A$

Anonymous

@BalarkaSen Yes. It makes sense

Anonymous

I get it now

Balarka Sen

This is nice

Anonymous

18:20

Follow the steps in the picture above

Balarka Sen

Well I have not seen the statement of the singular value decomposition yet, let me see that

Anonymous

Rotate-Scale-Rotate basically

Balarka Sen

So I can write any matrix $A$ as $U \Sigma V^T$ where $U, V$ are orthogonal and $\Sigma$ is diagonal?

That sounds fair.

I can prove this

The diagonal entries of $\Sigma$ are going to be the singular values, of course

Say $S$ is the unit sphere in $\Bbb R^n$ centered at the origin. Then $AS$ is the ellipse with semi-axes being $\{v_1, \cdots, v_n\}$ with $\|v_i\| = \sigma_i$ precisely the singular values

This basis of semi-axis vectors is an orthogonal basis

Let $V^T$ be the rotation of the orthonormal basis $\{e_1, \cdots, e_n\}$ to $\{v_1/\|v_1\|, \cdots, v_n/\|v_n\|\}$

Let $\Sigma$ be the "squishing" operation which squishes the unit sphere to the coordinate ellipse with semi-axes of length $\sigma_i$

Anonymous

Hmm, and $U$ is again a rotation operation

Balarka Sen

Yeah it should rotate the finished ellipse back to a coordinate ellipse

Anonymous

18:29

It's a good geometric analogy

Balarka Sen

Well, I'm a little scared. Why shouldn't $U = V$?

vzn

Eigendecomposition of a matrix

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. == Fundamental theory of matrix eigenvectors and eigenvalues == A (non-zero) vector v of dimension N is an eigenvector of a square (N×N) matrix A if it satisfies the linear equation A v = λ v ...

Balarka Sen

Oh nevermind, I see why

Anonymous

Well, why do you think so? Even if they were symmetric $V=V^{*}$ with real entries

Anonymous

I don't see why U=V

Balarka Sen

18:35

@Blue Yeah I had a wrong picture stuck in my mind. I was thinking you'd turn the same amount of angle you went from the standard basis to $\{v_1, \cdots, v_n\}$ as you would from going from $\{v_1, \cdots, v_n\}$ to $\{Ae_1, \cdots, Ae_n\}$. No reason that should hold...

Anonymous

@BalarkaSen Yep, the angles will be different!

Anonymous

Did you see the gif on Wiki?

Balarka Sen

I did not

Anonymous

Balarka Sen

good picture lol

Anonymous

18:37

Eek

Anonymous

Hehe

Anonymous

Click on it

Balarka Sen

yeah done clicked

Good gif.

Ok, so singular value decomposition is something very simple. OK

Anonymous

Yeah, it looks quite simple now. They have proved it using Spectral theorem in Axler

Anonymous

Although that's equivalent to your line of thought

Balarka Sen

18:41

Yeah

Anonymous

I need to learn how to cut and shape metal pipes now. Machine workshop tomorrow!

Balarka Sen

$A^T A$ is symmetric, is the point

So you can spectral theorem the shit out of it

@Blue Ugh

Anonymous

It's a good skill to learn. I can make weapons to keep humans away from me, now :P

Anonymous

LATHE MACHINE THREADING

►

Anonymous

We work on these ^

Anonymous

18:44

Well, I should go now...XD

Cows

My brain has done its thing. It has shifted into (pure )mathematics mode. I am officially in another universe now although my body inhabits this one. Cheers!! I will be in this mode for a while now.

Balarka Sen

@Blue good luck making machines

skullpatrol

@dmckee you're not alone.

Tell me what a "variable" is really?

Cows

Let's just enjoy life. The masters are no more τ(n) – Ramanujan tau function

0celo7

19:03

@BalarkaSen

Balarka Sen

"I am the shit like I slithered in poop" - Hopsin

0celo7

@BalarkaSen brings a tear to my eye

Balarka Sen

It truly does

Cows

@0celo7 what's up? What have you been up to recently?

@BalarkaSen hey dude

Balarka Sen

hey

Sir Cumference

19:18

@loocsieulb Gonna sound weird, but do you have those UToronto calc 1 notes?

Anonymous

19:29

@SirCumference Seems to be online: home.tykenho.com/LectureNotes137_Preview.pdf

Anonymous

math.toronto.edu/~alfonso/137/137.html

loocsieulb

@SirCumference why is it weird

didnt i send them to you?

blue linked it^

the first link

bolbteppa

20:02

My god, my god, Gelfand has a section on polynomial matrices in his linear algebra book (which I never even looked at) which proves the Smith normal form (without calling it that) and then re-proves the JNF again

Those pesky...

0celo7

20:22

@BalarkaSen fuck me, reading some old essays I wrote

r/iamverysmart material

0celo7

21:33

oi, I need a TeX god

bolbteppa

General SUSY algebra in $n$ dimensions with central charges, pure insanity even starting from Minkowski space, how does anyone know this stuff

Imagine what the general superconformal algebra looks like

vzn

22:21

← surfing/ looking into "law of maximum entropy production," finding lots of wild stuff o_O

Secret

23:18

nature.com/articles/s41467-017-02507-y

I think we are capable to simulate any particle with the quasi particular properties of atoms in these simulators

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