Mathematics - 2015-07-01 (page 3 of 4)

@Chris'ssistheartist since $$\frac{x-\log(x+1)}{x(x+1)\log^2(x+1)}$$ is integrable, the integral would be $\frac1e-\frac1\pi$ times that integral, no? Wait, it is bad at $0$

Owatch

No big deal. I just glanced at it now and wondered.

Hippalectryon

5:16 PM

Well, you definitely don't need to store it

robjohn

@Chris'ssistheartist yeah... I was being careless

Chris's sis the artist

@robjohn hehe, no pb :-)

@Hippalectryon @robjohn I think to add the generalization to my book.

Hippalectryon

Yep :D

Chris's sis the artist

@Hippalectryon Yeah, it's a nice integral. :-)

Owatch

@Hippalectryon Bad news

Hippalectryon

5:19 PM

@Owatch ?

Owatch

I put it in my sim.

Gravity is inversed

Everything repels each other!

Maybe I am not saving stuff right. Hold on.

Hippalectryon

@Owatch Aha I know. Did you kind of copy my code ? Pygame uses a reversed y axis (downwards)

Owatch

Yes.

I went through and made a conversion.

Hippalectryon

Does i even repel on the X axis ?

Owatch

It repels on x axis, yes.

Hippalectryon

5:22 PM

O_o w0t

Works fine on mine

Owatch

Give me a minute to look through it..

I added the moon and the earth in mine

On about the same x axis.

And it just started pushing the moon away

Hippalectryon

@Owatch Here's the latest version pastebin.com/DQE7WUrj

Owatch

What's new? Just more planets?

Hippalectryon

and minor fixes

Nothing that should influence the attraction though

Owatch

gravity has a - sign

Oh oops

Hippalectryon

5:25 PM

Yes it does. The force is attractive

It's along - the direction vector

Owatch

Well. I might know why mine was repelling

I just need to fix something with the way I save stuff first

(I make a copy of object array, but save to the copy) Need to change that

Hippalectryon

ok

oh lol

Owatch

I'm just not sure about the isEqual selector I use.

is A equal to a copy of itsself?

Or identical to a copy of itself

I believe I should use the latter.

Hippalectryon

Why do you need the isEqual operator in the first place ?

Owatch

Avoid calculating force on itself

Nah, isEqual works. Just checked. Cool

Hippalectryon

5:29 PM

Well I don't need that since I just check if the indices are the same, but if your way works it's fine too :-)

Owatch

I am using objects.

Hippalectryon

It's not that different

Chris's sis the artist

@robjohn how does that problem seem to you aesthetically? I'd like to add to my book its generalization.

Hippalectryon

I just place my objects ("items") in a table

That way by knowing their indices I know directly who they are

It also allows me to create separate tables easily, of objects needn't interact for some reasons

Owatch

Most of objective C is using objects

robjohn

5:31 PM

@Chris'ssistheartist It is nice aesthetically. You could just substitute $a,b$ for $e,\pi$

Owatch

I could do what you do, but I would need to map them to a dictionary inside an array or do some nonsense

Chris's sis the artist

@robjohn Yes. :-)

Owatch

Since all the sprites are objects to begin with, I just add properties to the class for position and velocity

Hippalectryon

Do as you like :-) everyone codes differently

Owatch

Is there a distance at which the proximity of an object to another is too large to allow for an orbit?

I tried adding an orbit at close range, and I got more of a slingshot effect

Hippalectryon

5:33 PM

@Owatch Nope in theory

However, as distance get smaller, errors grow bigger

Owatch

There is no friction. Hmm

Hippalectryon

Maths aren't perfect in computers

Owatch

I see

Hippalectryon

I even had to add an exception for when the distance is null because it's too small for the computer

Owatch

Isn't velocity vertlet especially bad for close range interactions?

Hippalectryon

5:34 PM

No, it doesn't matter as long as the precision (computer's prec + deltat) is good enough

Owatch

0_0

I added two moons

One collided with Earth, the other managed to escape.

Hippalectryon

What configuration exactly ?

Owatch

What do you mean? The setup in terms of masses and stuff?

Hippalectryon

yep. mass, distance

and angle

(initially)

Owatch

I don't know the angle.

Hippalectryon

5:36 PM

Well you can give an average

Owatch

It sets it up where ever to tap.

Let me check mass

Hippalectryon

The two moons will interact with each other so knowing where they are is important

Owatch

Earth is 50x heavier than the moons

The moons are the same mass.

And did interact a lot]

The orbit of the first moon was fine until the second one came around and threw it off

Hippalectryon

Came around ? The should be moving at roughly the same speed

But as I said, it depends a lot (!!) on their starting position

For instance if the two start roughly at the same position, they'll ignore the earth

They'll just wander wherever

Owatch

One was closer than the other, and instead of maintaining a fast low orbit, sharply curved near the far side of the earth from where it spawned, and then arced out in a large ellipse

Then it kind of pulled the second moon away

Hippalectryon

5:39 PM

Yeah, that sounds normal to me. Let me try that with my engine.

I'll have to add the possibility of setting the initial angle for that

Owatch

I have a far lower timestep than you

And much higher G constant

T = 0.010

G = 500 or so

Hippalectryon

O_o

Why not use the real values ?

Owatch

Stuff moved too slow?

I can increase time step

But does it matter if G is a constant ?

Hippalectryon

It does matter if you want to model real things (like the solar system)

And then compare with experimental data

Owatch

This is a more like a sandbox for kiddies.

But I can make G lower and tilmestep larger

If you have an iPhone, you can get early access to the application.

Hippalectryon

5:42 PM

I only have an old phone :(

I suppose there's some free emulator ?

Owatch

I don't have one at all.

But I test on the Xcode simulator.

There is not an emulator, since it is an app on my computer I am developing.

Hippalectryon

Ok I set the two moons opposite to each other

They orbit fine

Let's try with 4 moons :D

Owatch

Can you make a moon orbit a planet, that orbits the sun?

Hippalectryon

I'll try

Still orbiting fine :D

Now let's set a moon orbiting around a moon around earth

Owatch

In my simulator. I often add a large planet, then a small one with a moon

The larger one always pulls the moon away from the smaller planet

robjohn

5:47 PM

@Hippalectryon what is that?

Owatch

It is a shrine

Hippalectryon

@robjohn n bodies problem simulator

I give it planets and stuff and it simulates their movement

Pretty basic for now

robjohn

@Hippalectryon They are all just standing still...

Hippalectryon

@robjohn It's a screenshot

Owatch

He is trying to achieve this

Hippalectryon

5:48 PM

Haha

robjohn

@Owatch Ah, now it becomes clear.

Hippalectryon

@Owatch !!!!! yesss !

Ted Shifrin

hi @Owatch, @Hippa, @robjohn. @Stan: You did have it right at one point, but that file apparently didn't reflect your update. :)

Hippalectryon

@TedShifrin o/

Owatch

@Hippalectryon Did you achieve it?

Hippalectryon

5:51 PM

@Owatch It eventually loses its orbit, but that's most probably because at that scale the distances are rather small, and thus the deltat is too big

Owatch

Maybe it is distances and stuff to. My distances are small

Hippalectryon

But at the start it orbits nicely

Owatch

I added 4 moons, but they didn't orbit at all

Hippalectryon

@Owatch Try my settings

Owatch

They all plunged towards earth and then shot out again.

Hippalectryon

5:51 PM

O_o

Owatch

Could be a bug

With the way my stuff works

Random Variable

@Chris'ssistheartist I wrote up an evaluation for that integral involving $\text{arccot}(x)$ that you posted a while ago and that r9m mentioned yesterday. It uses $\log(1-e^{2ix})$. If you're interested, I'll post it.

Hippalectryon

You didn't manage to get pygame working (did you ?)

Owatch

I did

Hippalectryon

Otherwise you could try my program with your settings

Owatch

5:52 PM

Yeah, I can do that.

For some reason, you can set distances whereas I can't

Chris's sis the artist

@RandomVariable Sure, I'd be glad to see it. :-)

Hippalectryon

@Owatch code for the 4 moons pastebin.com/u44Bbfkz

Owatch

For example, you have the distance of your screen to be very large

But my distances are all dictated in points or pixels

Hippalectryon

Well that's why I define a scaleX and a scaleY

For that simul the size of my box is 2e9 meters

Then I convert those to actual pixels

Owatch

Is that a feature of pygame?

(Can it easily set that up for you I mean)

And do the conversions

Hippalectryon

5:55 PM

No I just implemented that, it's rather easy

For instance if you have a point at (x,y)

Owatch

Oh yes, I see

Hippalectryon

you'd just do x * scaleX, y * scaleY (assuming you don't need to recenter it, which I do since the center is in the upper left corner)

Owatch

Maybe I should do the same.

About how much 'actual' distance is your scale covering?

Ted Shifrin

hi, bananas

Hippalectryon

For now (the earth thingy) it's 2e9 meters

For the solar system I'm around 6e11 meters

Owatch

5:58 PM

Ok.

I've noticed that over time, the moons lose their pattern

There is also a fourth object coming in from above

Hippalectryon

@Owatch because verlet isn't perfect, nor the computer. Orbits are always unstable.

@Owatch In the version I gave you, when you click you spawn a new nameless moon

Owatch

The object has no name

robjohn

@Owatch Are you sure it is periodic?

Owatch

No.

Oh ok.

I thought you had put in a rogue moon of some sort

just to mess with me

Hippalectryon

@Owatch :P

@Owatch I'm working on changing that to work a bit like yours : when you click, it will pause the simulation and allow you to add new planets in orbit

By dragging and clicking

Owatch

6:03 PM

verynice

Hippalectryon

Never used pygame seriously before though xD I'll have to look up some tutorial

Owatch

I have a varied selection of objects

Hippalectryon

@Owatch is the emulator heavy (to download) ? because my connection is very bad

Owatch

Mine?

Hippalectryon

the iphone app emulator

Owatch

6:05 PM

You would need to have a Mac, or be running OS X. Then you would need Xcode.

Hippalectryon

Uh I'm on windows

Owatch

It's doesn't do much more than yours

Hippalectryon

But it's pretty :P

Owatch

Just fancy graphics and stuff.

Yeah. You cannot use it on Windows.

And how can your connection be bad!

You are in Paris.

Hippalectryon

Hotspot wifi

Owatch

6:07 PM

Although I admit that when I had Orange for wireless, they weren't the best..

Hippalectryon

(some kind of boarding school)

And even at home it's pretty limited (I live in the outskirts, right next to Paris, but the cable installation is old)

Owatch

Infrastructure needs an upgrade.

You don't mind if I use your method instead of mine in my App do you?

It's shorter than mine, and has less unnecessary stuff in it.

Hippalectryon

@Owatch Use my code any way you want :-)

I just request 584% of the profit this app will generate :3

Owatch

Thanks

Hippalectryon

My code is nearly always open source :P

Owatch

6:13 PM

It will probably generate none.

Because people keep complaining it doesn't do what they want

Hippalectryon

hehe

Owatch

But talking to you, it seems a lot of it is beyond my control.

Hippalectryon

Don't worry i'm here to help :D

"a lot" is probably exaggerated though

Owatch

Ok maybe.

Hippalectryon

Remember that orbitals are, by nature, unstable positions

Which meas that the slightest (inevitable) mistake will put the planet off its orbit

Chris's sis the artist

6:18 PM

@RandomVariable nice evaluation. :-) Also @r9m says it has one but he didn't post it yet.

Hippalectryon

The goal is to have the new orbit still look like a circular orbit by minimizing errors

Random Variable

@Chris'ssistheartist Thanks. I think he said he converted it to a triple integral. I'm not sure what he did.

Chris's sis the artist

@RandomVariable Neither do I. Hope he posts his solution.

Balarka Sen

not asleep yet, @Soham?

Soham Chowdhury

@Ted, you'll probably know: what subject studies things like Mobius transformations? (Not complex analysis, I mean the symmetry-ish bits only.) Group theory?

@BalarkaSen almost.

I just logged in to ask this.

Balarka Sen

6:25 PM

what d'you mean by symmetry-ish bit?

isometry groups, etc, are concrete group theoretic topics.

(plus, it's in Artin)

Soham Chowdhury

I was doing a few problems from Engel's book. They were on "transformation geometry". Amazing, amazing proofs of things like Napoleon's theorem. Around three lines each, and very illuminating.

The proofs almost feel like cheating.

Balarka Sen

yes, Klein's Erlangen Programme.

Soham Chowdhury

I googled around and rediscovered Mobius transformations.

Balarka Sen

It's all group theory.

Soham Chowdhury

@BalarkaSen "a geometry is the study of invariants under certain operations" :)

Balarka Sen

6:27 PM

nods

Soham Chowdhury

@BalarkaSen mmm, cool. Artin has a chapter on "Symmetry", I've seen.

Balarka Sen

yes. it has very cool stuff.

Soham Chowdhury

Aren't Mobius transforms related to . . . $\mathrm{PSL}_2(\Bbb C)$ or something similar?

I don't know yet.

Balarka Sen

it's an action by a matrix of PSL_2(C), sure, if you will.

when you learn about group actions, that'll be quite apparent.

Soham Chowdhury

ah.

anyway. thanks.

good night. :)

Balarka Sen

6:29 PM

ps : "symmetry of things" isn't actually quite a specific branch.

a lot of things comes up from it

for example, symmetries in the hyperbolic space throws up things like modular forms, etc.

g'night @Soham

Ted Shifrin

@Soham: Projective geometry ...

Balarka Sen

surely not just projective geometry, @Ted.

Ted Shifrin

I didn't say just.

Balarka Sen

oh, fair enough.

misinterpretation of "..."

Ted Shifrin

Sentence fragments are often vague :)

Balarka Sen

6:37 PM

@TedShifrin indeed ...

:P

Tobias Kildetoft

@TedShifrin Hi

Ted Shifrin

Tobias, just the person I wanted!

What's the Krull dimension of $\Bbb C[x,y,z]/(xz,yz)$? :)

Oh, and hi :)

Tobias Kildetoft

hmm

Ted Shifrin

It came up because of this question.

I have little intuition for non-equidimensional dimensions :P

Tobias Kildetoft

me neither. So obviously it is at most $3$, but I suppose it is also at most $2$ since it is a proper quotient (any chain in the polynomial ring with length $3$ includes the $0$ ideal)

Ted Shifrin

6:42 PM

I'm not convinced it's even at most 3.

Tobias Kildetoft

@TedShifrin Any chain of prime ideals will correspond to a chain of prime ideals in the polynomial ring containg $(xy,yz)$

r9m

@RandomVariable The $\displaystyle \int_0^\infty \frac{\cos 2nx}{(1+x^2)^2}\,dx$ part .. is were I needed to use double integral .. although I did it in an earlier step so I was dealing with a triple integral in one sense :) your approach is Awesome!!! :D (+1)

Ted Shifrin

But there are zero divisors, which throws me off.

I see, that's the reverse inequality.

Tobias Kildetoft

@TedShifrin yeah, I am having a hard time figuring out the prime ideals containing $(xy,yz)$

@TedShifrin what do you mean the reverse?

Ted Shifrin

I guess I'm worrying about a lower bound, not an upper bound.

Random Variable

6:45 PM

@r9m What was your starting point?

Tobias Kildetoft

@TedShifrin yeah, I am trying to get a decent upper bound so I have an idea of how many prime ideals to look for

Ted Shifrin

Plus, if we localize at $0$, how are the two Krull dimensions related?

(I mean the point $0$.)

So localize at $(x,y,z)$.

Tobias Kildetoft

@TedShifrin Hmm. Been too long since I calculated Krull dimensions

Ted Shifrin

A lot less long than for me :)

r9m

@RandomVariable converting $\displaystyle (1-x\tan^{-1} \frac{1}{x})$ into $\displaystyle \int_0^1 \frac{t^2}{x^2+t^2}\,dt$

Ted Shifrin

6:47 PM

@anon, @pedro: Have you any ideas?

When we worked on singularities, we often did lengths, but not Krull dimensions. But even that is 20 years ago. :)

Tobias Kildetoft

@TedShifrin I need to run again. I will think about it and let you know if I get an idea

Ted Shifrin

OK, thanks, @Tobias. We'll see if anon or pedro can do it :)

Mike Miller

you have a lower bound of two. are you expecting three?

Ted Shifrin

I was actually hoping for something smaller than 2. See the question I linked above, @MikeM.

Mike Miller

$(z) \subset (x,z) \subset (x,y,z)$

Ted Shifrin

6:50 PM

I don't see necessarily that those give prime ideals when you descend.

That's just up in the polynomial ring, sure.

Lucio D

Hi @DanielFischer

@TedShifrin Hi

Ted Shifrin

hi @Lucio

Lucio D

@TedShifrin How goes it?

Ted Shifrin

Indeed, @MikeM, it seems $(x) + (xz)$ is not prime in $\Bbb C[x,z]/(xz)$, right?

Chris's sis the artist

@r9m it's not bad using that $$\int_0^{\infty} \frac{\log(\sin^2(t/2))}{a^2+t^2} \ dt=\frac{\pi}{a}\log\left(\frac{1-e^{-a}}{2}\right) $$

Ted Shifrin

6:53 PM

Well, @Lucio, I don't get to eat all day, so I've had better days ... :) How're you?

Mike Miller

they're prime to start with. primes in $R/J$ are exactly the same as primes in $R$ that contain $J$. this does not require that $J$ is prime.

Lucio D

Sounds rough. Quite hungry myself. @TedShifrin

Ted Shifrin

Yes, @MikeM, of course i know that, but I'm getting myself bolloxed up.

r9m

@Chris'ssistheartist !!!!! :D Caught me by the neck! :-)

Mike Miller

do you agree, then, with what I said before?

Random Variable

6:55 PM

@r9m I was tentative to post it because I was worried my approach was similar to yours. Fortunately it wasn't.

Chris's sis the artist

@r9m :D

Ted Shifrin

Interesting, @MikeM, so are you telling me we get the same answers for $\Bbb C[x,y,z]/(xz,yz)$ and $\Bbb C[x,y,z]/(xz)$?

So maybe I want to get the union of two lines, rather than the union of a line and a plane.

Lucio D

@TedShifrin I watched a documentary about Vietnam today. There was a incident where the Viet Cong managed to find the USA army radio frequency and either got an American (or talented VIetnamese) to order a bombing on a certain location. So in effect, they were able to get Americas tanks to bomb their own infantry.

Mike Miller

@TedShifrin: Why shouldn't $\mathbb C[x,y,z]/(xy,yz)$ be 2-dimensional? That's the dimension of the largest variety it contains.

Ted Shifrin

I don't remember ever knowing about that back in the day, @Lucio, but maybe it was well known.

r9m

6:59 PM

@RandomVariable please never be :-) having different approach lets me learn new stuff .. having similar approaches boosts my ego (in a way) :D

Ted Shifrin

I wanted it to be $2$-dimensional, @MikeM, in fact. For the question I linked.

Lucio D

@TedShifrin Oh okay.

Mike Miller

this is the answer you're getting. you told me earlier you wanted the dimension to be smaller than that. I don't understand what you want anymore.

robjohn

@Chris'ssistheartist Do you use the series expansion for $\log(\sin(x))$ for that?

r9m

@Chris'ssistheartist Now what kind of MONSTROSITY is that?!!! :O :D

Lucio D

7:00 PM

@TedShifrin Will you be voting for Trump?

Mike Miller

Indeed we're done as Tobias said, because since $R$ is a proper quotient of the polynomial ring, it has to have dimension at most 2.

Ted Shifrin

LOL, I'm confuzled. So do I have a counterexample to that question, @MikeM ?

Hell no, @Lucio. What gives you that idea?

Mike Miller

I'm not reading the question. I'm answering the question you asked me.

Lucio D

@TedShifrin Was just interested. Watching his election speech on youtube.

Ted Shifrin

Right, @MikeM, so my intuition was right. This is a counterexample.

Mike Miller

7:01 PM

Whatever you say, boss.

Ted Shifrin

@Lucio: I'm a wacko liberal from way back in the 60s.

Chris's sis the artist

@robjohn I have a proof somewhere but I don't remember if I used the expansion.

Ted Shifrin

@MikeM: I'm talking about this.

Lucio D

@TedShifrinOh Oh okay :)

Ted Shifrin

So the intersection is a point, and the sum of the codimensions is in fact 2.

Mike Miller

7:03 PM

I'm satisfied in not clicking. Whatever's going on in that question is your domain. That's all I was saying.

Chris's sis the artist

@r9m I created it today. :-)

robjohn

@Chris'ssistheartist just thinking how I'd approach it.

Ted Shifrin

Well, if you told me I was right, @MikeM, I'd post an answer :D

Chris's sis the artist

@robjohn let me know if you find a nice way.I think I used some multiple integrals.

r9m

@Chris'ssistheartist well I must say it looks intimidating! will sth like it be there in the book too?

Ted Shifrin

7:04 PM

@MikeM: You talk with Danny yet?

Chris's sis the artist

@r9m Yeah, it might be the point $ii).$ from a problem with 2 points.

Mike Miller

Yes, a bit. Today is Canada day so I'm doing homework before talking to him on Thursday.

Ted Shifrin

Tremendous. Do say hi for me :)

r9m

@Chris'ssistheartist 2 points?! you are grading the problems too? :D (by difficulty I suppose)

Chris's sis the artist

@r9m I mean the problem is consisted of 2 points, first you calculate something and then you finish this one.

@r9m Well, it's not that hard.

r9m

7:06 PM

@Chris'ssistheartist ah! okay! I misunderstood !

@Chris'ssistheartist okay! the sight itself was too much for my pothead!

Chris's sis the artist

@r9m lol, pothead? This is a new word to me. :-)))

r9m

@Chris'ssistheartist people who smoke cannabis are called potheads (informal use)

Chris's sis the artist

lollllllll

@r9m I didn't know that. Interesting. :-)))

r9m

@Chris'ssistheartist similarly people who do crystal meth are called methheads :P (not that I have done that kind of stuff :P)

Chris's sis the artist

@r9m :-)

Pedro Tamaroff

7:12 PM

And then there is the worst kind of them all, mathheads.

4

r9m

:P Lollll

Balarka Sen

hi @PedroTamaroff

Pedro Tamaroff

Hey there.

Chris's sis the artist

Back in 60-90 min.

r9m

Attempted Murder :P

Tobias Kildetoft

7:39 PM

@TedShifrin Back. I see you got the answer from Mike

Pedro Tamaroff

@TobiasKildetoft Hello, Tobias.

OH, DRATS. It's late.

Tobias Kildetoft

@PedroTamaroff Hi

Pedro Tamaroff

I have to run. =/

We'll be talking later...!

Tobias Kildetoft

Man, I was really encouraged by my blog post getting nice comments. Then I just noticed that most of them are just there to link to various companies (via the name of the commenter) in an attempt to boost their search rankings

Nick

7:55 PM

Can anyone point me to a proof of the volume formula for an ellipsoid that uses synthetic methods? Or even just tell me what such a proof would look like

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$Mathematics$ Mathematics