The h Bar

Sir Cumference

00:41

Anyone else finding that SE's mobile links sometimes don't work?

Or just me?

danielunderwood

00:54

I think I've noticed some odd things like the app opening but not following the whole link

Sir Cumference

@danielunderwood I mean the mobile site

Links just reload the page

Akash. B

05:42

hi

Slereah

06:45

Back at work

RIP me

Abcd

07:06

Why is $\Delta G = 0$ at equilibrium?

@Secret If you have any idea about it, please reply. Thanks.

Secret

This is because it means there is no preference between reactants and products, thus there should not be an energy cost moving either way

which is exactly what thermodynamic equilibrium is about

John Rennie

@Abcd the Gibbs free energy is the chemical analogue of potential energy in physics.

So just like with potential energy the stable point of a system is a minimum in its potential energy i.e. a minimum in its Gibbs free energy.

Abcd

07:33

@JohnRennie but minimum $\ne 0$

@Secret Could you please provide a mathematical view?

Akash. B

hi

John Rennie

08:15

@Abcd at a minimum $dG/dx = 0$ where $x$ is whatever variable $G$ is a function of. So $dG = 0$.

That is, for a system at equilibrium if you change it by an infinitesimal amount in any way then the associated free energy change is zero.

Secret

08:32

Pretty much what JohnRennie said: at the minimum, the function is stationary, thus any infinitesimal displacement is constant hence dG = 0

Akash. B

hmm

Novalium Company

13:50

When we say 'electrode', we just mean something that is a good conductor of electricity? (such as copper, silver, aluminium...)?

Yashas

you can have a hydrogen electrode

Standard hydrogen electrode

The Standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be 4.44 ± 0.02 V at 25 °C, but to form a basis for comparison with all other electrode reactions, hydrogen's standard electrode potential (E0) is declared to be zero volts at all temperatures. Potentials of any other electrodes are compared with that of the standard hydrogen electrode at the same temperature. Hydrogen electrode is based on the redox half cell: 2 H+(aq) + 2 e− → H2(g)This redox...

Novalium Company

Hm, ok. And when we say 'electrolyte', we mean a substance which when dissolved in water (or other polar solvent), makes that solution electrically conductive?

@Yashas The hydrogen electrode thing seems complicated. I just want to understand the main meaning of the word electrode as used in physics or chemistry.

Yashas

I am not sure about what the scientific definition of an electrode is.

while hydrogen itself isn't a good conductor of electricity, a hydrogen electrode is

the hydrogen electrode contains a metal though

I am not sure about this now

Novalium Company

@Yashas Ok, no problems, thanks :-)

Anonymous

14:15

@NovaliumCompany It's explained here

Anonymous

@Yashas Hi! How's uni life treating you? :D

Yashas

@Blue I have a quiz on algorithms tmrow.

Assignment deadlines on Wednesday and Thrusday. One more on Sunday and another on Monday.

Exams from next ~Wednesday

Anonymous

@Yashas Sounds hectic. Similar here. We have tons of labs and the associated boredom of report writing

Anonymous

There aren't many upcoming exams though :P

Akash. B

15:23

hello

Anonymous

15:45

4

A: What is the intuition behind convolution?

No doubt there are many links and expositions on wikipedia and elsewhere about convolution, its relation to linear systems, and its link to frequency analysis, however I do not think that is what you are after. Forget convolution for a moment. Just think of nothing but averages: If I give you ...

Anonymous

Hmm, I'm not being able to link the notion of sliding averages as presented in this answer to this:

Anonymous

Anonymous

Any idea anyone? Pretty much stuck on this

Yellow

"In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps."

This is from wikipedia. How do you show the dot product as a tensor?

ACuriousMind

@Blue A sliding average is the convolution of the function you're averaging over with a rectangle function normalized to have unit area..

@Yellow The matrix representation of the standard dot product is the identity matrix. For any symmetric positive-definite matrix (= tensor of rank 2) $A$, $x^T A y$ defines a dot (or inner) product where $x^T$ is the row vector to the column vector $x$.

Anonymous

15:56

@ACuriousMind Thanks, but I'm not sure that sentence is completely clear to me. Let's begin with the diagram I posted. Say $f$ is the rectangle function with a unit area. You mean, $(f*g)(t)$ at some time, say $t = 5s$ is the "time average" value of the function $g(t)$ at time $t=5s$? How are we defining "average" in this context?

ACuriousMind

@Blue You're averaging the function over an interval whose width is the width of the rectangle.

So you should think of a window with that width sliding in from the left. The value of the convolution at a certain point is the average of the function inside that window when the window is centered at that point

Anonymous

@ACuriousMind Aha, I see! But how does this notion of sliding averages extend to cases where neither $f$ nor $g$ is a unit area rectangle?

ACuriousMind

@Blue When one of the functions is normalized to have unit integral, then you can call it a weighted sliding average.

If neither is normalized, the interpretation doesn't really extend.

For example, when you think of convolving with a Gaußian, that's a weighted average that counts points closer to a point much more strongly than those far away

enumaris

just got the power to edit people's posts...THE POWER...IT IS OVERWHELMING

Anonymous

@ACuriousMind Interesting! Getting the idea now. Well, as far as the "normalizing to get unit area" , that normalizing factor would always be a scalar (if we consider "energy signals" which we normally deal with in signal processing). So that scalar can always be taken out the integral and in that way, I think the intuition can still be extended?

ACuriousMind

16:04

@Blue Sure

Semiclassical

I have a certain amount of intuition for simple convolutions, e.g. the sliding average example

Anonymous

@ACuriousMind But one second, does a convolution of non-integrable signals even make sense?

Anonymous

Like when the area of one of them is not finite

Anonymous

Or neither of them has a finite area

Semiclassical

for generic convolutions, though, I have a much harder time getting useful intuition

Anonymous

16:05

(for "power signals" for instance...they don't have finite area)

enumaris

it doesn't help that in deep learning a Convolution is totally different

tangentially related

Semiclassical

yikes

there's also discrete convolution, of course

and that one does make sense as "product of power series"

(which in turn goes towards the "transform of a convolution is the product of the transforms" fact)

enumaris

fun beans

Anonymous

@enumaris I found something related: colah.github.io/posts/2014-07-Understanding-Convolutions

Anonymous

Haven't read about CNNs before though

enumaris

16:09

yeah the two different convolutions are related

but they aren't the same

a convolution in deep learning doesn't have the flipping the "time" axis part

and it's also discrete

but it does have the multiply then add the values together part

ACuriousMind

@Blue No, not generally

But of course you can get lucky: E.g. convolving a function with finite integral with a constant function certainly works

Anonymous

@enumaris From what I read so far, I don't think "time" axis is anything special in that regard. It's just like any other variable. In discrete convolutions like in image processing there's another dummy variable which we flip $(f*g)[n] = \sum_{m=-\infty}^{+\infty}f[m]g[n-m]$.

enumaris

as far as I'm aware, there is no flipping of the kernel

I'm not familiar with your notation as it pertains to deep learning

Anonymous

Umm, what's the definition of convolution in deep learning ?

Anonymous

@ACuriousMind Makes sense :)

enumaris

16:15

but if $g$ is the kernel, then no you don't bother to flip the $g$ since you are learning the kernel anyways.

the colah blog does a good job showing what it is a bit further down

the convolution in DL is more like a cross correlation...except without the displaced time axis either lol

or maybe the time axis is there...not sure -.-

It's hard to explain the convolution in DL in words only lol

need to draw a picture...

but once you draw a picture, it's super simple

is it just me or is anna v's answers all a bit...weird...

(maybe not all, since I haven't seen all of them)

Anonymous

Convolution vs Cross Correlation

►

Anonymous

It does seem to be that they're looping in the reverse direction for convolution

Anonymous

Which is essentially similar to reversal of the time axis

Anonymous

When you do the convolution the kernel gets automatically flipped. So there's no need to manually flip it and then find the cross correlation

Anonymous

16:27

(I think so)

Anonymous

machinelearninguru.com/computer_vision/basics/convolution/…

Anonymous

"You can find a list of most common kernels here. As previously mentioned, each kernel has a specific task to do and the sharpen kernel accentuate edges but with the cost of adding noise to those area of the image which colors are changing gradually. The output of image convolution is calculated as follows:

Flip the kernel both horizontally and vertically. As our selected kernel is symetric, the flipped kernel is equal to the original.
Put the first element of the kernel at every pixel of the image (element of the image matrix). Then each element of the kernel will stand on top of an eleme…

Anonymous

"
but if g
is the kernel, then no you don't bother to flip the g since you are learning the kernel anyways."

Anonymous

I don't know what that means

Secret

Convolution is basically: Let the distributions be made of sticks. Next you scale whatever target distribution with the height of the stick, repeat for all the sticks and then add them all together

This is why expanding polynomial expression is a discrete convolution

Adirian

16:33

Does anybody know what I should look at, if I wanted to calculate the potential energy of really unusual electron orbits?

Secret

what do you mean by unusual?

Anonymous

@enumaris The "time axis" (which replaced by $\tau$) in this situation is in essence the $(u,v)$ axis as in $$G[i,j] = \sum_{-k}^{+k}\sum_{-k}^{+k} h[u,v]F[i-u,j-v]$$. This is similar to $$\int_{-\infty}^{+\infty}f(\tau)g(t-\tau) d\tau$$

Anonymous

The only case we wouldn't have to do the horizontal and vertical flips for image is when the kernel is symmetric

Anonymous

But I don't see any reason why it should be symmetric in general

enumaris

the kernel is learned

Anonymous

16:36

@enumaris What does that mean? :/

Adirian

(Specifically, I am trying to figure out how much, if any, force is applied between two protons of arbitrary distance, given an electron is orbiting one, from the potential set of orbits in which it is circling both, independent of the electrical forces themselves.)

Anonymous

You mean it uses some kind of data sets as inputs

enumaris

@Blue the convolutions in DL aren't done by set kernels. The kernels are set to random initial values and then learned via back prop

Anonymous

To adjust its parameters?

enumaris

The "weights" that you have in standard dense neural nets are replaced by kernel values

Adirian

16:38

So by "unusual" I mean "How much energy is tied up in an imaginary/potential orbit between two protons some distance apart, if any" - a long-distance ionic bond, sort of?

Anonymous

Okay, in that case it makes sense, since while learning it creates its own matrix via some slight nudges produced in the right direction by the datasets. But in that case, that matrix shoudn't be called the kernel at all! The horizontal and vertically flipped version of it should be called the "kernel" to be technically correct

Anonymous

So it's basically a semantics issue? :P

enumaris

I suppose if you index the kernel from -k to k rather than from 0 to k you get a form that looks like a convolution lol

but no coding language indexes from -k to k...it's always from 0 to k

Anonymous

You call the flipped version of the actual kernel as the "kernel" in DL?

Anonymous

Good to know :D

enumaris

16:40

There's no need to flip

since the kernel is not fixed

it's learned

so what's the point of having a randomly initalized kernel, then flipping the initializations...

Anonymous

@enumaris I got that part. There is no need to flip the matrix. But what I'm saying is that that matrix shouldn't technically be called a "kernel" in that case.

enumaris

and yet it's called a kernel :D

Anonymous

For convenience's sake, that sounds fair enough :)

enumaris

in any case, if you go into DL with the knowledge of continuous convolutions like I did, you do get quite confused on what the hell a "convolutional neural network" means

lol

Anonymous

I guess to make it accessible to the general public they had to create their own terminologies for a lot of things

danielunderwood

16:42

sounds like quite a convoluted topic

Anonymous

@enumaris Yeah, I can see

Anonymous

:P

enumaris

I kept thinking that you were gonna convolve the image with itself

but that didn't seem like "learning" to me

danielunderwood

Speaking of NNs, do you ever have layers that aren't fully connected aside from dropout? And do people use different cost functions?

Anonymous

98

Q: A list of cost functions used in neural networks, alongside applications

What are common cost functions used in evaluating the performance of neural networks? Details (feel free to skip the rest of this question, my intent here is simply to provide clarification on notation that answers may use to help them be more understandable to the general reader) I think it w...

Anonymous

16:50

@danielunderwood Yes, convolution neural nets aren't fully connected

Anonymous

They're only connected to the local neurons (in the previous layer)

Anonymous

Which is quite evident from the image sliding over another image example I guess :P

enumaris

17:03

yeah

generally "fully connected NN" or "Densely connected NN" only refers to the vanilla neural networks

CNNs and RNNs don't fall into that category

you can also have stuff like variational autoencoders which use a sampling layer

Secret

@Adirian I don't think you can do classical treatments like orbits, you have to deal with them quantum mechanically which requires some computational chemistry calculations of the binding energy of these electrons

Anonymous

Hmmm, I'm finding this signal processing class to be pretty enlightening XD These stuff seem to be applicable to a wide range of fields. Also for the first time I'm seeing engineering profs worrying about stuff like integrability, which is quite nice :P

Anonymous

(Gives the much needed relief from the semiconductor devices classes ;_;....where they've been boring us with transistors for one whole year)

Emilio Pisanty

17:18

lolz

are they seriously surprised that users ignore that header?

have they even been to the internet?

have they been somehow time-warped from the 1990s?

Anonymous

"I profess my own inability to understand this situation. Did we overshoot human perception? Did we make it so noticeable, so.. obvious, that it could not be seen from within; Like humanity itself being unaware of the entirety of the universe around them?"

Secret

Banner blindness

Banner blindness is a phenomenon in web usability where visitors to a website consciously or subconsciously ignore banner-like information, which can also be called ad blindness or banner noise. The term "banner blindness" was coined by Benway and Lane as a result of website usability tests where a majority of the test subjects either consciously or unconsciously ignored information that was presented in banners. The information that was overlooked included both external advertisement banners and internal navigational banners, often called "quick links." Banners have developed to become one of...

you cannot er... combat against subconscious bais

ACuriousMind

18:20

@EmilioPisanty I recently looked over the shoulder of someone browsing without an ad blocker, and I was struck by how full their screen was. Which led me to speculate that designers and programmers who have only ever used adblockers for years tend to forget how trained the average internet user is to ignore flashy colorful banners.

Secret

NB: I don't ignore banners, I just don't click on them lol

Novalium Company

18:37

Guys, what exactly is a charge? Is it like a physical object that has mass or like a soul of something (e.g electron). And also if an electron is the smallest charge carier, then wht doesn't it have 1C of charge, won't that be convenient?

Semiclassical

Now I find myself wondering how old the Coulomb as a unit is

Anonymous

@NovaliumCompany Charge is more like a "property" or "description" of a system in simple words rather than a physical object

Semiclassical

@NovaliumCompany historically, the Coulomb as a unit of electric charge is obtained from the Ampere as a unit of electric current, as Coulomb = Ampere * Second

Novalium Company

So when we say something has charge, we mean that it has the ability to attract/repel other charge carriers?

ACuriousMind

@NovaliumCompany Yes.

Semiclassical

18:42

and in the original 1881 definition of SI units, one has an ampere defined as "A tenth of the electromagnetic CGS unit of current. The [CGS] electromagnetic unit of current is that current, flowing in an arc 1 cm long of a circle 1 cm in radius, that creates a field of one oersted at the centre"

ACuriousMind

Also, the electron is not the smallest charge carrier, quarks have a fraction of an electron's charge

Semiclassical

and an oersted in turn has some other definition that I don't remember

Novalium Company

Hmm, my book lies then :D

Semiclassical

point is that the definition of an Ampere (and therefore a Coulomb = Ampere * Second) is extracted not from electrons but from a tabletop setup

So there's no reason to expect Coulombs to relate particularly well to a microscopic charge

Novalium Company

Hmm, got it.

Semiclassical

18:45

Also, I referenced above that SI units were originally defined in 1881

by contrast, the electron wasn't discovered until 1897

Novalium Company

When was first defined, the ampere or coulomb?

Semiclassical

That's a good question and what I was wondering about above.

Novalium Company

I mean, you cant define the ampere, before knowing what coulomb is?

Semiclassical

In SI units, the Coulomb is a derived unit whereas the Ampere is a base unit

So in that scheme the Ampere is defined first and then the Coulomb is defined from it

Anonymous

@NovaliumCompany You could define ampere in terms of force exerted

Semiclassical

18:48

Yeah. That's the modern definition

the historical definition seems to be a bit more obscure

Anonymous

Doesn't seem too interesting anyway :P

Semiclassical

10 Amperes = the electromagnetic CGS unit of current = "that current, flowing in an arc 1cm long of a circle 1cm in radius, that creates a field of one oersted at the centre"

Novalium Company

1C = 1A * 1s I mean, you can't define the coulomb that way if the ampere wasn't defined before that?

Semiclassical

so historically the ampere was defined in terms of the magnetic field you could produce with that current

and then the Coulomb would presumably be defined in terms of how much charge that current would transport in one second

Novalium Company

^ yep

Anonymous

18:52

I think it is much more natural that in the past people observed the effects of current first. Also, it is much easier to accurately measure current than charge

Anonymous

Which again is a reason that A is considered to be one of the 7 base SI units

Semiclassical

The other basic point is that SI units were designed to be useful for everyday use, i.e. typical experiments in that area of history

Anonymous

And not Coulomb

Anonymous

@Semiclassical Yep, exactly

Semiclassical

and even today that's still true: an electrician typically worries about amps, not coulombs

of course, it depends on the discipline. a chemist would care about electron charge far more than electron current

and that's why you'll tend to see units like the electron-volt in place of the Joule

(Where things really get confusing is magnetic field units. oersted vs. tesla vs. gauss, sheesh)

Anonymous

18:56

Yes, but I don't think chemists are concerned with precise measurements of electron charge. They tend to deal with $e$ as an unit of charge in itself without worrying about its value and I guess that was true for a long time

Semiclassical

@Blue that's true

Anonymous

But of course we had the oil drop experiments and stuff later on

Semiclassical

they do care about having integer multiples of the electron charge, tho

redox reactions etc

so in that context the electron charge is itself a meaningful unit of charge, regardless of what its value in SI units would be

Anonymous

Right, yup

Anonymous

As far as explaining chemical reactions go, that basic model goes a long long way

Novalium Company

18:58

Hmm, why charges in a conductor spread at the surface?

Semiclassical

units are a human invention, and therefore different disciplines can have different conditions for what's useful

Novalium Company

and do they flow at the surface?

Semiclassical

imagine you've got two free electrons nearby in a conductor. What will they tend to do?

Novalium Company

repel

Semiclassical

yep

more generally, they'll move into locations which minimize the potential energy

and it turns out that the best way for free electrons in conductor to minimize the potential energy is to spread out on the surface

Novalium Company

19:02

Hmm... interesting. I thought spreading them in the whole conductor would be best at minimizing the electrostatic force.

Anonymous

@NovaliumCompany If they spread out homogenously on the surface then w.r.t a single charge the other charges are placed more or less symmetrically and that minimizes the net electrostatic force (on individual charges) too

Semiclassical

That might be a good calculation, then. Suppose you've got two spheres of equal charge, but with one you take the electrons to be concentrated on the surface and on the other to be spread out homogeneously

What you should be able to show is that the former will require less work to assemble than the latter

(Note: I don't remember how hard it is to do this computation.)

Anonymous

Should be easier on a plane metal plate I suppose

Novalium Company

Hmm, ok then. But they don't flow (when emf is present) on the surface of course?

enumaris

weez

Anonymous

19:08

0

Q: Godels Incompleteness Theorem(s) and Consciousness is Non Algorithmic

This past March, when I called Penrose in Oxford, he explained that his interest in consciousness goes back to his discovery of Gödel’s incompleteness theorem while he was a graduate student at Cambridge. Gödel’s theorem, you may recall, shows that certain claims in mathematics are true but ca...

quantum-mechanics soft-question mathematics biophysics biology

Semiclassical

ugh, consciousness

Anonymous

I wonder why they tagged that as QM

enumaris

consciousnessnessnessnessness

everything top of the line is quantum

haven't you seen Ant Man 2?

Anonymous

lol

Novalium Company

Omg enumaris just saw it 5 hours ago!!!

Semiclassical

19:10

@Blue well, it's not entirely off base: en.wikipedia.org/wiki/Quantum_mind#Penrose_and_Hameroff

Novalium Company

enumaris I think our minds are entangled.

Semiclassical

of course, it'd be nice if that was actually evident in the question itself

Anonymous

@Semiclassical Oh, nice link

Anonymous

I hear a lot of people saying that Penrose's work in that area is a bit crackpotish but I don't know how far that is true

Semiclassical

eh, it's less crackpot than some directions at least

The direction I dislike is "consciousness intervenes on quantum processes"

Penrose's direction is more "quantum processes directly influence consciousness"

Anonymous

19:13

Yes, that one mainly :P

Semiclassical

penrose's direction may or may not be correct, but I think it's at least defensible in a way that the other direction isn't

Of course, even that direction has more-or-less defensible variants

enumaris

@NovaliumCompany the quantum chaos makes our minds quantumly entangled quantum

Novalium Company

@enumaris You forgot one more 'quantum'.

Semiclassical

even if one believes that QM and consciousness are inherently related, that doesn't imply telekinesis/telepathy

enumaris

it might imply quantum telekinesis and quantum telepathy

Semiclassical

19:16

the predictions made by QM remain statistical in nature

enumaris

you mean the quantum predictions made by QM remain quantumly statistical in quantum nature?

Semiclassical

...

yyyy do u hurt me

Novalium Company

:D

danielunderwood

@Blue just what I was looking for. Evidently your google skills are much better than mine

enumaris

:D

danielunderwood

19:18

:E

Semiclassical

$:\exists$

Anonymous

Anyhow, diverting a little bit. I was looking for areas in numerical computation which could benefit from a speed-up due to quick solutions to high dimensional linear equation systems. Finite element method for PDEs looks like one such area. I'm essentially looking for possible applications of the HHL algorithm.

Anonymous

But one catch is that the solution to that algorithm only prepares a state $|x\rangle$ which encodes the actual solution $\vec{x}$. Reading all the components of $|x\rangle$ would take $\mathcal{O}(N)$ time, which makes the exponential speedup due to the HHL useless. So I'd prefer problems which involve high dimensional linear systems but only require reading of a few components from the final solution

danielunderwood

texmojis could be the next big thing

enumaris

anyone here familiar with dependency parsing?

danielunderwood

19:22

like python requirements files/install_requires?

Anonymous

arxiv.org/abs/1010.2745

Anonymous

This looks somewhat relevant

enumaris

natural language dependency parsing

I know in spirit what it does, but I don't know the details that well

bolbteppa

When $\Gamma_{\mu \rho}^{\nu}$ is not symmetric, should it be $\nabla_{\mu} A^{\nu} = \partial_{\mu} A^{\nu} + \Gamma_{\mu \rho}^{\nu} A^{\rho}$ or $\nabla_{\mu} A^{\nu} = \partial_{\mu} A^{\nu} + \Gamma_{\rho \mu}^{\nu} A^{\rho}$?

enumaris

start from $\nabla_\mu A_\nu = \partial_\mu A_\nu -\Gamma^\rho_{\mu\nu}A_\rho$ and raise indices

Novalium Company

19:30

What was that 'state of matter' called when the substance is between the liquid and solid state? (e.g icecream, yoghurt)

enumaris

It's undergoing a phase transition, so it's simply a mixture of the 2 states

there's no name for it beyond that afaik

Novalium Company

@enumaris sliqoid? :D

enumaris

quantum sliquid

Semiclassical

presumably, if you look at a small enough portion of that state, you'll find that it's either in the liquid or the solid state

Novalium Company

@enumaris xD

enumaris

19:35

hmm

danielunderwood

oh nope. I can tell you all about installing python packages though!

lol

enumaris

hmmm

danielunderwood

pip install and you're done!

enumaris

conda install also works for some subset of packages

Novalium Company

(if you have pip...)

enumaris

19:37

assuming you have anaconda

like me

danielunderwood

conda heathens!

enumaris

lol

bolbteppa

@enumaris you sure it's not $\nabla_\mu A_\nu = \partial_\mu A_\nu -\Gamma^\rho_{\nu\mu}A_\rho$?

$\nabla_\mu A_\nu = \partial_\mu A_\nu -\Gamma^\rho_{\mu\nu}A_\rho$ is memorable but so is $A_{\nu ~ ; ~ \mu} = A_{\nu~ ,~ \mu} - \Gamma_{\nu \mu}^{\rho} A_{\rho}$ :(

enumaris

hmmm

bolbteppa

I'm happy to not care but I'd say @0celo7 would cry

(Unless he went back to engineering from lack of math stimulation :( )

enumaris

19:51

I think it would be a definition then...

what you can prove is that any two derivative operators satisfying several properties differ at most by $(\nabla'_\mu-\nabla_\mu)t_\nu = C^\rho_{\mu\nu}t_\rho$

if the connection is not torsion free, then your definition is arbitrary based on what you call $\nabla$ and what you call $\nabla'$

I'm not sure tho

I don't work with connections with torsion lol

bolbteppa

Yeah I am not sure if it's convention or some legit issue people may ignore

enumaris

I also wonder if the proof that $\Gamma$ is the unique differential operator that is compatible with the metric breaks down if torsion is allowed

so perhaps you can have multiple connections with torsion that are all compatible with a metric...

not sure...

user54826

20:14

hello people, I've found a user who copies and paste the same answer to a variety of questions, but he modifies here and there to adapt it

as a result, his answers are utterly long and repetitive. he could just mention he already wrote a very similar answer and that the case applies, instead of a wall of copied and pasted text

but that's funny as hell if you ask me :D

enumaris

if it's only minor modifications, he should probably just flag the question as a duplicate

and link to the other question

danielunderwood

Is it a legit answer and not an ad or rant?

user54826

nah the questions are sometimes very different

I do not know @danielunderwood many of his answers get negative points, but then again, his most popular one is a copy and paste of his own answer that got negative votes

it's like, he always manage to put that same wall of text on very different questions, it "somehow fits"

that's funny as hell, but man, I'm tired to see his new answers when they are like that. I skip the whole wall of text

that's 1 funny case for sure, I had never seen such a case before

I can even google his username with 3 words of that wall of text and a pack of his answers are recovered

he's been doing that since at least 2 years now

I opened 9 tabs with his same copy/paste answer. google indicates more but I stop here

enumaris

reading about topology in Wald's appendix a....very concise...but not very intuitive

user54826

makes me wonder about a robot that would copy and paste John Rennie's answer on ALL relativity questions. many such answers would get downvotes, but some would get surprisingly many upvtes

despite being exactly the same words lol

enumaris

21:00

ugh

need some more intuitive sense for these point set topology definitions...

bolbteppa

Yeah it's not a nice appendix

I wonder do people who study topology know it's about approximation

I wouldn't get that sense from a topology book, no no...

enumaris

I feel like Appendix A in Wald is good for someone who already knows point set topology and just needs a refresher on the theorems in it

bolbteppa

Yeah he doesn't prove anything so it's more an indication of what you should vaguely know I think

enumaris

I'm fine with no proofs

but I would like an intuitive notion of those concepts

and how the definition actually fits in with the intuitive notion

like how does the definition of compactness have anything to do with what I would think of as "compact"

bolbteppa

I've spent a few years trying to find that intuition

enumaris

21:06

Or how does the topological definition of continuous corresponds to the usual definition...he just says that it's easy to see that it does in the case of R lol

hmmm

bolbteppa

It took me a long long time to find out what the f is going on with compactness

enumaris

heh

I just think of it as closed and bounded subsets of R^n...cus meh...

bolbteppa

If everything in topology is about approximation, how the hell could compactness even really matter, who cares if a space is 'compact' in the usual sense if we know things are 'near' or 'far' from one another using the topology...

enumaris

o.o

I'm not familiar with the "everything in topology is about approximation" viewpoint

bolbteppa

"That's it — a topological space is just a set of points together with a "nearness" relation that satisfies these axioms" math.stackexchange.com/a/1795952/82615

Basically we think of 'nearness' with real numbers, but bananas are nearer to apples than to microwaves etc

Bourbaki topology intro gives the best 1 page explanation of this for a reference, surprising given how abstract the actual book is

You can set up a 'nearness structure' on a set of points from multiple perspectives, from open sets, from neighborhoods, from closed sets, from limit points, from bases, all different ways of setting up a structure one can approximate with

enumaris

21:18

so you're equating nearness as "is contained in the same open set"?

oh wait, you referred to a totally different definition of a topological space

let me read that post lol

bolbteppa

Basically, any element of an open set is 'near' to any other element in that set, that is, any element of an open set can be used to approximate (i.e. stand in for) any other element in that open set up to an error which quantifies membership of that open set

enumaris

but the whole set itself is an open set..

I think the post specifies only points "near" sets

rather than points "near" other points

But his definition has no "not near"...so I'm a little confused

like if $P\notin A$ then is $P$ not near $A$?

bolbteppa

I think that post defined a topological space in terms of a base

Base (topology)

In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies are most easily defined in terms of a base which generates them. == Definition and basic properties == A base is a collection B of subsets of X satisfying these two properties: The base elements cover X. Let B1, B2 be...

So you can take a base as your primitive notion and then define everything in terms of them

Wald takes open sets as the primitive notion and defines a base in terms of it

So you need to describe the definition of a topology in terms of open sets via nearness in a direct way without focusing on points and their membership, i.e. in a global instead of local sense

enumaris

hmmm...

the way that the post is written

I can easily make the (I think mistaken) assumption that "nearness" is the same as "is in"

bolbteppa

No, that's how you're supposed to do it

enumaris

21:32

wat

bolbteppa

4

Q: What is a topology?

Having read through the mathematical definition of endowing a set with a topology I must admit that I'm still struggling to conceptualise what such a mathematical construct is. I've read articles that talk about a topology on a set as defining a notion of "nearness" between elements of that set ...

general-topology differential-geometry

enumaris

if $P$ is near $A$ if and only if $P\in A$ then how does "nearness" give any new information beyond $\in$?

bolbteppa

A finite set example is better than thinking in terms of $R^n$

ACuriousMind

@bolbteppa No, it essentially defines them in terms of a closure operator: "All points near $A$" is simply the closure of $A$ in the tradtional nomenclature.

That is "$P$ is near $A$" is just a different name for "$P$ is in the closure of $A$".

enumaris

hmmm that would make sense I suppose

ACuriousMind

21:36

The axioms are just "The empty set is closed", "A set is contained in its closure", "The closure of a union is the union of the closures" and "If $P$ is in the closure of $A$ and $A$ is in the closure of $B$, then $P$ is in the closure of $B$".

enumaris

and if we intuitively think of closures as "nearness" then we can intuitively understand topology better? o.o

ACuriousMind

@enumaris Well, in the usual topology on $\mathbb{R}$ (or any metric space), taking the closure effectively means adding all points infinitesimally close to a set in terms of metric distance, so it's not that far-fetched to use as intuition.

enumaris

so how is compactness defined using this language?

bolbteppa

I'm not sure if you can call this perspective the Kuratowski closure axiom perspective exactly, I think the post is more taking a base perspective, usually if you want to talk about points AND closed sets you define limit points

enumaris

cus the "every open cover of A admits a finite subcover" definition has no intuitive meaning to me

bolbteppa

21:41

But ultimately it's the same thing

enumaris

ah crap

bolbteppa

It's very frustrating trying to really nail this stuff down

I recommend thinking of the idea of compactness as coming from trying to tame divergent sequences (or nets/filters if you want to be general)

enumaris

Wald's shows the exterior derivative is unique by using the symmetric property of the connection...-.-

bolbteppa

Lets say you have a given sequence/quantity, how do you know if it converges? In general, you wont, so maybe you can assume it diverges and then end up showing it doesn't, compactness seems to come from that thinking

enumaris

now I gotta wonder how exterior derivatives, which should be well defined without any notion of a connection, actually should be defined...-.-

hmmm

bolbteppa

21:49

Lets say you take a convergent sequence, and you append to it a bunch of blips (say every 10'th element is $10^n$ that diverge off to infinity, we 'know' it's really just a convergent sequence, plus some stuff we want to ignore... That's the whole limit point thing, calling a sequence which at least has a subsequence which converges to have a limit point... I can go on a rant to make it obvious, but basically Pugh defines a compact set as one for which every sequence has a convergent subsequence

(and I can go on another rant to link this to open sets if needed)

ACuriousMind

@enumaris Finite intersection property: If I have a infinite collection of closed subsets such that all finite intersections between them are non-empty, then their total intersection is non-empty. In the nearness language: For an infinite collection of sets such that for finitely many of them there is at least one common point near them, there is least one point near all of them.

If you stare at this long enough, and especially if you look at the counterexamples, you realize it means your space has no "far horizon" to which one could escape.

bolbteppa

The question is, when can you always be sure that even if you have (a) divergent (sequence) sequences, we will always be able to use (it) them to approximate at least something...

enumaris

@ACuriousMind I see...

bolbteppa

@ACuriousMind that's great

enumaris

hmmm, I just got told I have a limited knowledge of ML...wonder what parts I'm missing -.- Cus I thought I had at least a pretty high level understanding of most parts of ML...

I guess it's hard to know what I don't know :D

unknown unknowns so to speak

bolbteppa

21:56

I am too busy with the known unknowns :(

The answers are somewhere north, south, east and west of my current understanding

enumaris

o.o

ACuriousMind

At least they're not down or up ;)

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