The h Bar - 2019-05-03

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10:39 AM

@JohnRennie Any thoughts on this question about the structure of the CMB?

I'm still not 100% sure about this. According to Wikipedia "in the ΛCDM model [...] objects at redshifts greater than about 1.5 appear larger on the sky with increasing redshift". — PM 2Ring 1 hour ago

1 hour later…

11:53 AM

@PM2Ring makes sense to me. As space expands, not only does the distance from you to the source increase, but the distance between edges of the source increases too. The light beams become more separated and, as a result, the object appears to subtend a larger portion of the sky

2 hours later…

2:13 PM

Wow! Two positions and three candidates?! :)

user351417

I had never seen three simultaneous enter-chat animations before. It looks hideous.

2:40 PM

@Mo_ well there was a fourth. Lots of comments in chat about it & two questions on Meta.se, if you're interested in the details

1 hour later…

3:58 PM

@kylecampbell maybe have a look at the representation theory of discrete groups as applied to molecular physics and solid state?

@peterh I'm still looking forward to an explanation of what this was. The conversation you replied to was about personal employment matters. I'm assuming that you had good reasons for butting into such a conversation.

I mean, other than the fact that @ACuriousMind linked you to a comment and you decided to disregard the context and push your own agenda, maybe?

still, it's pretty rude to leave my request for explanation unanswered.

Abhas Kumar Sinha

@PM2Ring Heheheheh XD LOL best, that's lit, very funny. Did he got the question he wanted atlast?

Does anyone know how to solve these kinds of differential equations like $$\ddot \theta = - \dfrac gl \sin \theta$$, where $\ddot \theta$ refers to Second Order Derivative wrt time. (This is the equation of simple pendulum), without approximating $\sin \theta$ as $\theta$

I heard that there is a technique called the Trial method that can help me here, but haven't found that over internet :(

4:23 PM

@EmilioPisanty I've put you also into my chat ignore list. This is the first time as I ignore someone on the whole SE. Please, try to ignore me if we meet on the main or the meta site, I will try to do the same. Bye.

user351417

Um. Do we need to star that kind of conversation?

@Rishi I now actually stopped it.

user351417

@peterh My mistake. I wanted to say star : P

Abhas Kumar Sinha

@peterh What happened? you seem disappointed?

@peterh Any impression of "hate" lives exclusively inside your mind; if that is how you choose to interpret constructive criticism that you do not like, then that's entirely your choice. (Characterizing requests for explanation, after you've butted into conversations about personal matters, as "attacks", on the other hand, is not OK.) I would rather have you sustain your points in open debate, but if you start conversations I do expect you to stick around for the responses.

4:26 PM

@AbhasKumarSinha Please, let talk about the black holes.

Abhas Kumar Sinha

@peterh What you want to talk about blackholes?

Novalium Company

I just read about light diffraction (where the slit or obsticle must be really tiny, preferrably close to the wavelength of the light) and I was thinking, when I put my hand away from an object, the shadow's edges get blurrier, right? But why do they get blurrier, shouldn't the shadow's edges be just a bit smaller, but not blurrier?

Abhas Kumar Sinha

@peterh Do you want some cool stuff about Gravity Time Dilations?

@NovaliumCompany Do you know how optical rays work? That's the answer

@AbhasKumarSinha It was just an example. What I am currently thinking on, if there are multiple time dimensions and they somehow don't lead to contradictions, then the Noether-theorem would also result, that there are multiple energies.

Abhas Kumar Sinha

@peterh Not, sure but if you have some cool, Graduate level question/research ideas, I'd refer you to www.physicsoverflow.com . I'm not an expert in this particular topic, so please don't take my arguments seriously. i hope that you understand that I'm still a High School kid :)

Novalium Company

4:29 PM

@AbhasKumarSinha I do, but that doesn't answer my misunderstanding.

@peterh That's entirely your choice. But to the extent that you're requesting that if you post unconstructive proposals on meta then I should give you a free pass, then no, that's not going to happen.

@NovaliumCompany that's because the light source has a finite size. The blurred area you're seeing is called the penumbra.

@AbhasKumarSinha I like PO, but I think I am few to become a worthy contributor there.

393

Q: How does light bend around my finger tip?

When I close one eye and put the tip of my finger near my open eye, it seems as if the light from the background image bends around my finger slightly, warping the image near the edges of my blurry finger tip. What causes this? Is it the heat from my finger that bends the light? Or the minuscule...

optics visible-light diffraction

Abhas Kumar Sinha

@NovaliumCompany Do you know tubelight? A source where more than one point acts as a source of light, so, when you draw the light-ray diagrams, you'll see that the points intersect at more than one point, that's the cause of blur. So, answering your question, you should verify this fact first.

4:32 PM

@NovaliumCompany dscussed here:

12

Q: What determines the sharpness of a shadow?

What are the factors that affect the sharpness of a shadow? I would think that the distance between the light source and the object, the distance between the object and the shadow, and the size of the light source would all have an effect. How do they affect the shadow exactly? What is the fu...

visible-light photons everyday-life geometric-optics shadow

Abhas Kumar Sinha

@NovaliumCompany Oh sorry, I misunderstood your question. :P I thought, that was about multiple lights source :P

@peterh No problem, you can become a worthy contributor.

Novalium Company

@AbhasKumarSinha np :P

@peterh For the record, responding to a question by insultingly alleging that others are not using "their rational mind" when they've been polite to you in turn is not acceptable.

Abhas Kumar Sinha

@ACuriousMind Can you answer mine please? :P

@AbhasKumarSinha Your what?

Abhas Kumar Sinha

4:35 PM

question

Sorry, about three different conversations simultaneously are a bit hard to follow ;)

Abhas Kumar Sinha

18 mins ago, by Abhas Kumar Sinha

Does anyone know how to solve these kinds of differential equations like $$\ddot \theta = - \dfrac gl \sin \theta$$, where $\ddot \theta$ refers to Second Order Derivative wrt time. (This is the equation of simple pendulum), without approximating $\sin \theta$ as $\theta$

@rob try again

1743

Q: FAQ for Stack Exchange sites

Community FAQ For sites in the Stack Exchange 2.0 network To see a list of commonly used words and phrases, see the glossary. For official guidance from Stack Exchange, visit the Help Center. Asking questions How do I write a good title? How can I get answers fast? Where can I ask a ques...

now links to the election explainer

Abhas Kumar Sinha

@ACuriousMind What do you think? Can it be done?

@AbhasKumarSinha that equation cannot be solved exactly in terms of elementary functions

Abhas Kumar Sinha

4:39 PM

Why?

it can be solved in terms of a family of special functions called incomplete elliptic integrals

but it can be shown that those integrals cannot be evaluated in terms of elementary function

Abhas Kumar Sinha

never heard about that...

@EmilioPisanty show

@AbhasKumarSinha There is no general procedure, and all but the simplest equations do not have closed-form solutions for any reasonable definition of "closed form". This is one of these cases - you can write down some series/integral expression that solves it , but that expression could as well be just the symbol that denotes the solution to the equation - it doesn't really clarify anything to write it down

Wikipedia has the formula, if you want to see it

@AbhasKumarSinha "why?" is, in general, a difficult (impossible) question to answer, but in this case your question mostly reflects expectations that don't match reality

In general, integrals of elementary functions cannot be evaluated in terms of elementary functions.

In general, differential equations involving elementary functions as coefficients cannot be solved in terms of elementary functions

The cases that can be solved in terms of elementary functions are exceptions, not the other way around.

Abhas Kumar Sinha

@EmilioPisanty So, what's the best thing I can do here?

4:42 PM

so the core answer to "why can't this be solved?" is "because that's the default behaviour"

@AbhasKumarSinha it's called "reduce to quadratures"

Unfortunately we tend to teach mostly exceptions to this rule, so the misguided expectation is very understandable :P

i.e. reduce to a single integral that forms an unsurpassable roadblock

Abhas Kumar Sinha

@ACuriousMind cough cough A high school grader too XD

and then basically what we do is define that integral as a brand-new function which acts as a black box where we put all the stuff we couldn't solve to elementary form

in this case, it's these:

Abhas Kumar Sinha

@EmilioPisanty Can you link some sources, so that I can learn more from them?

4:44 PM

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form f ( x ) = ∫ c x R ( t , P ...

@AbhasKumarSinha Do a search for "The AGM Simple Pendulum" by MB Villarino. The AGM is the arithmetic-geometric mean. Gauss found that it's connected to elliptic integrals. There's a simple AGM formula for the exact period of a simple pendulum, but I'm not sure if it can be used to create a function for the pendulum's full cycle of motion.

@AbhasKumarSinha I'm not sure any of the resources I can link to will be appropriate for your level

it depends on how strong your current calculus background is

Abhas Kumar Sinha

@EmilioPisanty Oh okay no problem :)

it may be reasonable for you to have a go at a textbook on differential equations

Abhas Kumar Sinha

@PM2Ring I've to learn elementary integrals (not done yet!) before going to elliptic ones...

Well, Can time period is possible without Equation of pendulum? arxiv.org/pdf/physics/0510206.pdf

4:47 PM

@AbhasKumarSinha oh, then this is definitely something you need to put on a shelf and come back to at a later date

@AbhasKumarSinha In that case, definitely don't worry about elliptic integrals yet!

Abhas Kumar Sinha

@EmilioPisanty To an estimate how much time would I need to cover both?

@PM2Ring XD.. Was thinking same...

@AbhasKumarSinha If you're thinking that there is a binary classification for integrals such that any given integral is either 'elementary' or 'elliptic', then that is not the case.

"elementary integrals" could mean just the basics of integration

But if you can code, take a look at the pendulum period formula here: leapsecond.com/hsn2006/pendulum-period-agm.pdf

or the specific class of integrals of elementary functions that can be evaluated in terms of elementary functions

4:51 PM

That link comes from here: leapsecond.com/hsn2006 which has other fun stuff about precision pendulums & timekeeping.

Abhas Kumar Sinha

@EmilioPisanty Can expanding the sine function in the algebraic terms using the Taylor Series and using it upto the required accuracy will help me here?

@PM2Ring That looks interesting, I'll be making a 3D Photoreal Pendulum Simulation in Godot Game Engine for now XD :) :P

beyond that, there's just "integrals". There is an enormous array of special classes of integrals that cannot be solved in elementary form. Elliptic integrals are one such class. Other simple examples are Bessel integrals, exponential integrals, and Fresnel integrals -- though note that this is not an exhaustive list.

@AbhasKumarSinha help you to do what?

Abhas Kumar Sinha

@EmilioPisanty Using the sine expansion : Like $\sin \theta = \theta - \dfrac{\theta^3}{3!} + \dfrac{\theta^5}{5!} \dots$ and replacing it instead of $\sin \theta$ using the required number of terms (depending upon the required accuracy) can help me to get that differential equation done?

"get that differential equation done" is too vague of a description. There's many degrees and forms of what "solving" a differential equation can mean.

Abhas Kumar Sinha

Yes

@EmilioPisanty Will it help me in solving the integration?

4:59 PM

If you mean "If I expand the trigonometric function in the ODE to get $$\ddot \theta = -\frac{g}{l}(\theta -\tfrac{1}{3!} \theta^3),$$ will that allow me to get a closed-form solution of the differential equation in terms of elementary functions?", then the answer is no.

Abhas Kumar Sinha

okay...

the answer isn't to see how you can manipulate the equation into a "solvable" form

the answer is to step back, and to ask "what, exactly, do I want out of this differential equation?"

(and we cannot answer that question for you)

you want f(x)

@enumaris yes, but what does that mean?

do you want a closed-form expression, and that's it? (as I said, this does not exist in this case.)

it means a function of x

5:01 PM

@EmilioPisanty A sense of purpose and peace of mind

@enumaris ah, dammit, I thought you were OP.

(Does that qualify as a Lego Grad Student answer? :P )

please don't do that.

@ACuriousMind it's a start.

as in, it's only a start.

but I'll counter with

"Realizing that his attempt was a start but only a start, the grad student gives it up as a bad job."

(zzziiiiiinnnngggg)

Abhas Kumar Sinha

@EmilioPisanty So, arclength of integral can't be measured?

@AbhasKumarSinha what do you mean by "measured"?

5:05 PM

@AbhasKumarSinha Yes, of course you can exactly solve differential equations. For example, the exact solution for your equation is K(x).

Abhas Kumar Sinha

@EmilioPisanty discover the exact size of a quantity

@knzhou that's.... not helpful to how we're explaining this.

@AbhasKumarSinha I should mention the other half of the puzzle, though. In short: the fact that we cannot evaluate an integral (or a solution of a differential equation) in terms of elementary functions does not mean that we need to give up and that there's nothing else we can say about it.

(apologies for the over-emphasis, but it really is that important.)

@AbhasKumarSinha K(x), also known as the knzhou function, is a function defined to be the solution to that differential equation. If you want to know the values, I can numerically compute some for you.

If you think that's cheap, remember we do the exact same thing with the integral of 1/x. We make it a new function, and it's called "log".

(Of course you can define log in other ways, but for the purposes of this explanation...)

Abhas Kumar Sinha

@knzhou Good idea ;)

You feel okay with log(x) and not okay with knzhou(x) only because you know various rules that log(x) obeys and have a rough idea of its numerical values.

There's no inherent difference.

Abhas Kumar Sinha

5:09 PM

For an ellipse, $$\dfrac{dy}{dx} = \left ( 1- \dfrac{2x}{a^2} \right)\left ( \dfrac{b^2}{2y} \right) $$ and using it in the integral $$\int \sqrt{1+ \dfrac{dy}{dx}}dx$$ is the required integral length...

You can use a computer or pencil and paper to compute log(x), and you can do the same for knzhou(x). You can reduce other, more complicated integrals to ones involving log(x), and you can do the same for knzhou(x), and so on.

In the particular case of the pendulum, what we do is solve the ODE all the way to the cleanest integral we cannot go on from, and then put that integral into a box (which we might call "logarithm" or "incomplete elliptic integral of the first kind"), and then we study the living crap out of that integral

as it turns out, there's a lot we can say about integrals even when we cannot solve them in terms of elementary functions

Abhas Kumar Sinha

@EmilioPisanty Okay, Now I'm understanding the problem here...

for elliptic integrals, see the wikipedia link from before, or have a brief browse through the corresponding chapter in the Digital Library of Mathematical Functions to get a sense of the amount of stuff we're able to say about them

do you want the values? they're easy to calculate (say, using numerical integration, or other methods)

do you want the Taylor series? similarly easy

Abhas Kumar Sinha

@EmilioPisanty No, I'm I know it

5:12 PM

do you want their relationships with other families of special functions? it can take some doing, but it's done, and it doesn't require an explicit solution

@AbhasKumarSinha I was speaking generally

Abhas Kumar Sinha

Oh okay...

I got the point

more generally, I strongly recommend reading this:

Why are special functions special?, Michael Berry, Physics Today, April 2001, p. 11

Abhas Kumar Sinha

Can we solve all the elementary integrals? Or there are some which can't be integrated (ignore functions like Greatest Integer Function and modulus function stuff, here I'm asking for continuous functions only)

(both for Abhas and for anybody else here who hasn't read it.)

@AbhasKumarSinha what do you mean by "elementary integrals" there?

integrals of elementary functions?

Abhas Kumar Sinha

@EmilioPisanty Those integrals which involve elementary trig functions and algrbraic stuff..

I mean is there any shortcut to avoid manipulation and get the integral done

5:16 PM

No.

Abhas Kumar Sinha

Oh okay

there is an algorithm that is able to determine whether the integral of a given elementary function is an elementary function or not

Risch algorithm

In symbolic computation (or computer algebra), at the intersection of mathematics and computer science, the Risch algorithm is an algorithm for indefinite integration. It is used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch...

it is not for the faint of heart

it is too complex to be used by humans, really

Abhas Kumar Sinha

OMG!! XD

and it is certainly not an alternative to the usual methods

Abhas Kumar Sinha

okay..

Hey, this is not related..

5:18 PM

integration is just harder than differentiation, period

"Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.

$\uparrow$ that's a pretty accurate summary of the situation

Analytical integration, yes

Abhas Kumar Sinha

@EmilioPisanty I think that's more like trigonometry like manipulation ...

Amusingly, I find the reverse is true when it comes to numerical integration/differentiation

@AbhasKumarSinha sorry, I don't understand your comment.

There are way too many good xkcd's.

5:21 PM

Finding ellipse arc lengths is historically important, because orbits (in 2 body systems) are ellipses. So it was rather annoying to learn that calculating ellipse arcs in terms of the known elementary functions is a hard problem.

Abhas Kumar Sinha

@EmilioPisanty Like solving the trigonometry identity, one needs to command the trigonometrical manipulation, I think same needs for integration

I found something very funny, that even nonsense is mathematics XD : en.wikipedia.org/wiki/Abstract_nonsense

@AbhasKumarSinha there's a vast number of integrals, both solvable and not, where no amount of trigonometrical manipulations will help you.

Abhas Kumar Sinha

oh okay...

@AbhasKumarSinha When you learn about integration, that knowledge of trig functions will come in very handy. There are several important integrals where inverse trig functions pop up. It's easy to show why that's so by differentiation, but you wouldn't guess it just by looking at the integral.

@AbhasKumarSinha I can recommend this video series, though

Overview of differential equations | Chapter 1

Abhas Kumar Sinha

5:27 PM

@PM2Ring That's because that derivative of inverse trig functions are purely algebraical

@AbhasKumarSinha Correct.

@AbhasKumarSinha that's because the trigonometric functions are defined as the integrals of a certain class of algebraic functions

it's the same deal as with the logarithm

@EmilioPisanty That's so nice...that little pi is amazing. His channel is so good:-)

(as in: it's actually exactly the same type of integrals; the precise connection requires complex variables, but they're all one and the same.)

@user8718165 he's not actually a little pi

(as it turns out)

Abhas Kumar Sinha

@EmilioPisanty Hey, just as I use the taylor expansion to ding the limits, can same be used for Integration

5:30 PM

@AbhasKumarSinha it can. Sometimes it works, sometimes it doesn't.

Abhas Kumar Sinha

@PM2Ring Then how do you recognise them in bunch of random algebraic expressions?

oh okay

@AbhasKumarSinha practice, and experience.

@EmilioPisanty yes...he isn't:-)

Abhas Kumar Sinha

@EmilioPisanty oh okay...

If you already know the basics of differentiation and integration, then the way to get that practice and that experience is just to integrate without end

I used the Schaum book

basically, get a book that has a ton of exercises on integration (and which, of course, will only ask you to solve integrals that are actually solvable), and do all of them

5:38 PM

What Emilio said. In the mean time, when you're doing differentiation, keep your eyes open for cute patterns. They'll come in handy. Eg, differentiate ln(sec x + tan x)

Novalium Company

5:49 PM

Deep image reconstruction: Geometric shapes

Does this mean they can hook up the fMRI and can read what I'm imagining?

hmmmmm

biorxiv.org/content/10.1101/240317v2

so biorxiv abstracts allow hyperlinked references?

and biorxiv author lists allow ORCIDs?

man, they've one-upped arXiv for sure =|

except for the name :P

@ACuriousMind yeah, it's not one of the better brandings to come out of the past decade

biorxiv lul

6:54 PM

I guess it was inevitable that this would hit the HNQ: physics.stackexchange.com/questions/477602/…

7:12 PM

@PM2Ring If you think it shouldn't be there, please flag. We're still deciding how to use our remove-from-HNQ feature. We can discuss in here, but the flag is also helpful.

(If you think it's okay on HNQ, that's also fine.)

Unfortunately, I don't think my proposal of banning everything tagged quantum-interpretations from HNQ would fly :P

@rob I think it"s ok on the HNQ, and so far it hasn't gone crazy.

@ACuriousMind LOL. As soon as I see a psychoceramic answer there, I'll ping you. :D

@PM2Ring It's not the bad answers the questions attract. It's that even the best of them are still more metaphysics than physics in my eyes.

@ACuriousMind Fair point. Speaking of questions that will tend to attract metaphysical answers: physics.stackexchange.com/questions/477699/…

Ah, I see you've already commented there.

7:28 PM

Yup :P

It amazes me how often people think "reality" is a simple and obvious concept but quantum mechanics is not.

(I think it is magnitudes harder to have a remotely sound conception of what "real" means than to do quantum mechanics :P)

As the late Robin Williams said: "Reality...What a concept!"

thermomagnetic condensed boson

7:42 PM

@knzhou I invite you to write an answer to my question physics.stackexchange.com/questions/477602/… and/or let me know which answer is best so far

7:53 PM

@thermomagneticcondensedboson I'd say it's still too early to call. Give it another day or so.

But beware, scores may get inflated due to the HNQ effect.

thermomagnetic condensed boson

what's that HNQ? Hot new question?

2 out of three, it's Network, not New ;)

Those things from all sites you see in the side bar on the main site.

thermomagnetic condensed boson

I see

@thermomagneticcondensedboson Yes. That box on the bottom right. If a question gets into the HNQ, gets around 5-10000 views and a lot of upvotes. And also its answers.

thermomagnetic condensed boson

upvotes number depend greatly on whether the question is easily understandble to non physicist

my question about paper got 125 upvotes. but the one on bloch electrons only 24 votes. both went there

i suspect this one on QM and interpretation will not reach 24 votes

(i'd be surprised if it made it to 20 votes)

8:10 PM

QM interpretations are one of the most mainstream parts about QM I would say though, as someone who doesn't study QM at all. they're usually explained to non-physicists in a really pop-sci way

There's a current science fictional question. physics.stackexchange.com/questions/477704/… It doesn't break any laws of physics, though, just biology. So I guess it's not non-mainstream.

I find it fascinating how polarising QM interpretations are. "My interpretation is mostly satisfactory, it just has 1 or 2 problem areas. But your interpretation is patently absurd!" ;)

@JMac That's the worst thing about them! At their core, interpretations are metaphysical arguments about which parts of the formalism have ontological meaning, i.e. correspond to something that "really exists". It makes only sense to argue about them if one both knows the formalism and already has some overarching ontological beliefs about what "really existing" means.

@ACuriousMind Yup. Like how often you see people go along the lines of: "Well physics predicts multiple universes, so there must be a version of you that...."

"Can you somehow communicate with that universe?!"

and not often enough does someone point out that it was never what a physicist was saying

Due to the human need to tell ourselves stories, we tend to teach the stories and the ontologies they impose together with the formalism or even without it, fooling ourselves that being able to recite the story means that we "understand" it. And of course, for a certain value of "understanding", that is true. My pet peeve is people talking as if Feynman diagrams depicted any sort of "real" processes rather than a graphical encoding of an integral.

The problem with the stories is that they don't allow you to properly reason about the physics - at some point, you'll stretch the metaphor beyond its validity and be confused that the physical formalism does not predict what the extension of the metaphor would have.

When it comes up in the context of students learning classical physics, it probably doesn't help, because you usually have such concrete and clear examples to work with, whereas demonstrating QM isn't as straightforward as cause and effect

8:35 PM

@JMac Neither is Newtonian physics! Look up Norton's dome ;P

It demonstrates for a very simple case that Newtonian physics does not inherently possess the structure of "cause and effect"

Now, you may object that this just means the formalism oversimplifies the real world, but you may as well believe that for QM. Why one formalism leading to counter-intuitive results in some situations is acceptable and the other would not be is hard to explain

2 hours later…

10:52 PM

so, folks

Marble Race: MarbleLympics 2019 E5 - 5 Meter Sprint

any favourites for this year's MarbleLympics?

3

lol I haven't watched any of the recent ones, but those videos are really relaxing to me

@JMac yeah, I can't quite pin down why, but they're awesome

it also says a hell of a lot how the trappings of sports can get you rooting for inanimate glass spheres in no time at all

3

the combination of sports commentary, inanimate objects, and novel races is great

and the quality of it is way higher than you would expect for the concept

yeah, the production value is crazy

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