/sci/ - Science & Math

[Anonymous No.16191312]

Why can't some problems be solved analytically? What makes certain problems difficult or impossible to solve beyond numeric approximation?

[Anonymous No.16191337]

>>16191312
Can you name a single one though?

[Anonymous No.16191343]

>>16191337

[Anonymous No.16191349]

>>16191337
Well, I was principally thinking of those types differential equations that don't have exact solutions.

[Anonymous No.16191350]

>>16191337
Looks like nope.

[Anonymous No.16191355]

>>16191337
op bros...

[Anonymous No.16191361]

>>16191349
Name a single one

[Anonymous No.16191365]

>>16191312
Why can't some numbers be written as a fraction? What makes certain numbers difficult or impossible to write as a fraction beyond an approximation?

[Anonymous No.16191366]

>>16191365
inb4 the Wildebergers show up sperg out about only the rationals existing.

[Anonymous No.16191367]

>>16191361
Burger's equation or Navier stokes? I don't know why you're insisting on specific problems because I was asking why this happens in general.

[Anonymous No.16191370]

>>16191312
Usually it has to do with being non-linear. Nonlinear pdes don't have general solutions much like polynomials of degree 5 or more don't have general solutions. Although there's a proof for polynomials but there's no proof for pdes as far as I know

[Anonymous No.16191373]

>>16191366
Out of all the faggots on this board, those memelords are least of /sci/'s problems.

[Anonymous No.16191376]

>>16191373
This is true. I'd rather deal with them than a strict determinist or strict materialist (which are truthfully fairly similar as you pretty much have to be a strict materialist to believe in strict determinism without invoking some sort of God-like entity).

[Anonymous No.16191393]

>>16191366
You shut your mouth curr! Wildberger did nothing wrong!

[Anonymous No.16191396]

>>16191393
Horsefuckers? Is there a meme I missed?

[Anonymous No.16191412]

>>16191396
https://implyingrigged.info/wiki/2022_4chan_Autumn_Babby_Cup#Group_C

[Anonymous No.16191424]

>>16191412
Ah, so it's mainly a /vg/ thing? Okay. I've genuinely never noticed those threads anywhere and don't go on /vg/.

[Anonymous No.16191427]

>>16191370
So what generalizations can be made about problems with computational solutions in other fields?

[Anonymous No.16191457]

>>16191424
It's a 4chan wide event, /vg/ is just where the bulk discussion happens, though most boards try to make a gameday thread during an active tournament.
This was one of /sci/'s from last autumn >>/sci/thread/15851815#p15851815

[Anonymous No.16191496]

>>16191457
Last autumn I was probably too busy schizoposting in a determinism thread to notice.

[Anonymous No.16191607]

>>16191312
>>16191349
Because their differential equations describe properties that cannot be replicated using normal functions. You can define any arbitrary way to relate a function to its derivative(s) but the only way you will get an analytical solution is if those properties match that of some combination of elementary functions.

If I tell you y' + y = 0 it means the solution is one where the function's derivative is equal to the negative of itself, which is the defining trait of an exponential decay. However I can just make up some bullshit relation that y'''3 - log(arcsec(x*y'))*y" = x^y that shares no properties in common with any elementary functions, and you won't be able to get a solution to it beyond defining a new function as the solution.

[Anonymous No.16191669]

>>16191312
Skill issue

[Anonymous No.16191670]

>>16191607
>beyond defining a new function as the solution
But wouldn't that still be an exact solution?

[Anonymous No.16191674]

>>16191312
It all comes down to inverses. Some functions simply do not have inverses and thus cannot be linearized.

It's fairly common in fact. All computational methods, btw, simply utilize small, invertible or linear functions and then stitches shit back together.

[Anonymous No.16191677]

>>16191670
In a way, sure, but in practice, no.

If you can only describe it as 'the function that is the solution to X' that's not really feasible for actual use. Fundamentally it implies you have to use a computational method every time you want to evaluate the function of use it any meaningful way.

[Anonymous No.16191686]

>>16191674
>>16191677

[Anonymous No.16191799]

>>16191370
Aren't non-linear analytic equations representable as a Taylor series of polynomials? Meaning, each nonlinear function is implicitly a 5+ degree polynomial

[Anonymous No.16192090]

>>16191312
The difficulty of solving a problem rises exponentially with its dimensionality, so highly dimensional problems can only be solved through dimensionality reduction.

[Anonymous No.16192200]

>>16192090
Dimensionality reduction leads to losing information

[Anonymous No.16192397]

>>16191799
A Taylor series is an approximation using finite terms, its the power series that are exact/

[Anonymous No.16193585]

>>16192200
That wouldn't contradict my answer even if you could prove it.

[Anonymous No.16193714]

>>16193585
How so?

[Anonymous No.16193758]

>>16191376
I'm a finitist materialist. Now what?

[Anonymous No.16193817]

>>16193714
Because it would still be a viable way to approach problems that are too complex for an analytic solution, even if the solution can never be exact.

[Anonymous No.16193945]

Non-linear bros, what's our response?

[Barkon, King of Sci/Put your name as Knight of Barkon to worship No.16193960]

Fart on me

[Anonymous No.16194004]

>>16193960
Hmm, nyo.

[Anonymous No.16194042]

>>16193758
Idk man, continue to be retarded I guess?

[Anonymous No.16194054]

>>16194042
Imagine being an infinite believing non materialist. That's the real retard.

[Anonymous No.16194064]

>>16191312
Wait until you find out there are noncomputable PDEs.

[Anonymous No.16194083]

>>16194064
Enlighten me.

[Anonymous No.16194252]

>>16193945
Monte Carlo numerical approx for higher dim PDEs

[Anonymous No.16194367]

>>16194054
Let me know when you figure out what the "biggest number" is. Similarly, let me know when you figure out what materially constrains the truth value of "2+2=4."

[Anonymous No.16194875]

>>16194252
But that's still a computational method

[Anonymous No.16194877]

>>16194875
It's an approximation method. It's not an exact solution.

[Anonymous No.16194907]

>>16191337
You have two unit circles. How close do their center points need to be so that the overlap of the circles is exactly half of one circle's area?

[Anonymous No.16194916]

>>16191337
You have two unit circles. What is the distance between their center points if the circles overlap exactly half of each other's area?

[Anonymous No.16194930]

>>16194916
Is this a classic problem? I've never seen it before and it's unintuitive to me that there wouldn't be some polynomial of pi solution.

[Anonymous No.16194941]

>>16194930
Dunno how classic it is or isn't. But I'm not the first one to ever ask this problem. You can google the first few decimals of this problem's numerical solution and see that the problem shows up on google and on reddit and all kinds of places.

[Anonymous No.16195504]

>>16194877
Brainlet take. Solve it exactly

[Anonymous No.16196273]

>>16194916
>>16194930
looks like the distance would be 2sinx, where x is the solution to 2x + sin(2x) = 1/2 . If x were some multiple of pi, then sin(2x) is some fraction, but that would mean both 2x isn't and is rational - a contradiction. If x is a is a nonzero fraction, then obviously we get rational+irrational = rational , also impossible. x=0 doesn't fit the eq. Which means x is some non-pi irrational number.

At that point the problem is the same as >>16191365. You'd need a numerical method to approximate it.

I mean, honestly. Since pi is irrational, per OP's question you can't even write this out "analytically". The basic equation sin(x) = 1 has the solution pi/4. You need a numerical method to approximate it. We just give the solution the label pi because pi is so ubiquitous and important.

[Anonymous No.16196275]

>>16196273
pi/2 im sorry

[Anonymous No.16196337]

>>16195504
What is the brainlet take? I'm not saying that monte carlo methods are the only way or something. The point of >>16194875 was to point out that Monte Carlo methods are an approximate optimization approach.

They aren't like gradient descent where (if your function is locally convex around a minima in a region of measure greater than zero) you might actually get the exact analytical limit as your number of iterations goes to infinity. At best with an MC method you can get a computationally stable asymptotic average (which isn't the same as an analytical solution).

[Anonymous No.16196340]

>>16196337
The point of >>16194877 oops.

[Anonymous No.16196344]

>>16196273
This post just makes me wish I had the time to do numerical analysis in grad school.

From my understanding, most numerical analysis people don't consider e and π to be "not analytic" because we have closed form solutions to series which produce them exactly.

When I think of a non-analytic solution I think of something like a local minima to a convex non-quadratic function where there may in general be no closed form analytical solution, despite the convexity guaranteeing the existence of at least one proper local minima.

[Anonymous No.16196363]

>>16191337
Who should I marry? Should I get married? Is Becky or Sarah the girl for me? What should I do for a living? How much tax should which people pay? What should it be spent on?
Compute an answer to these problems, I implore you.

[Anonymous No.16197274]

>>16196337
Just use AI bro

[Anonymous No.16198159]

>>16196344
>don't consider e and π to be "not analytic"
Why would anyone even be consider then to be non-analytic at all?