Hey @Mike and @Juan!

@Ted Yeah, I find it plausibly torsion.

hi

How's everything going?

Hey guys! Hey @Mike

I'm kind of stuck on a question for a while: $min_x ||x||_{2,w} + \frac{1}{\alpha}||x-y||_2^2$

Since the weighted L2 norm is not differentiable, I have to make sure that $x \in d(||x||_{2,w})$

It's ok, @Daminark. Done with conferencing and going home tomorrow.

I can't really split this into single dimensions like the $||\dot||_1$ norm and solve for closed form :/

Some relationship between art and mathematics

Hey sorry @Mike, I had to go because I had my housing lottery

And that's good, hope you had fun!

Topology of tonality

Alternatively, would it be a migration candidate? It seems chemistry-focused enough to me to not be suitable, though.

@hBy2Py: I wouldn't object to it being migrated, but I agree that it seems like an odd fit. For instance, in my case, without knowing much about resonance structures I definitely feel like I couldn't answer the question in a way that lends itself to the long-term durability requested of an SE answer.

@EricStucky <nod>

Probably there are Chem.SE community members able to answer, but I at least would need some help with interpretation of the graph theory terminology.

@EricStucky so your solution is completely flawed? :p

Indeed.

so the entire statement is still needing a solution?

XD

No, the entire statement is false :P

I made an edit to my post

umm...

In all situations?

O.O

hmm, I think in all interesting situations, yes.

Ooookie, this math chat place is weird. I can tell you're making formalism jokes and I just. Don't. Get. It.

that makes no sense...

;-)

:/

For instance, if $y'=g(x)$ and $g$ is not continuous, you can get a continuous solution to the IDE which is not a solution to the ODE.

ok

erm, if the IDE has any solutions at all, of course.

erm

then explain y' = d(x)

where d is dirac delta?

Easy: that's not a function

ok fair enough. :P

what about heaviside?

pretty sure that's a common one to pop up in differential equations

hmm

Right, so it's kind of confusing

most intro DEs stuff doesn't really treat existence of solutions properly

oh

I mean, you'd probably say that

Yeah: I(y)=H(t) is solved by y=t*H(t)

in an intro course

but that's not differentiable, right?

to be fair I(y) isn't in an intro course

yeah, I see that isn't differentiable

Right, they'd just say y' :P

which is what I'm getting at here

which is then a paradox as we even HAD solutions that were not differentiable

^

I've never bothered to look into the details here

are people claiming things are solutions that are not solutions?

perhaps you can just say 'weak solution' and everything is fine

or is there something else going on?

weak solution? what's that mean?

Roughly speaking, any differential equation can be written instead as an integral equation; a solution to the integral equation is called a weak solution to the differential equation.

ah

and would the statement be true for 'weak solutions'?

or at least less trivial to claim it isn't true. :p

Yeah

I only know how to formalize weak solutions for linear ODE

(cause I see what you mean and by that basis almost anything involving a step function has no solution)

But in any case, I'm pretty sure that a weak solution is very weak and would be really shocked if it didn't cover IDE-type solutions

(yeah)

well a weak solution still needs to be continuous, right?

I mean a weak solution does not even have to be continnuous.

:P

O.O

well then what is the sort of solution that lower level differential equations courses tend to emphasize?

where solutions need to be continuous

but not necessarily differentiable?

:/

I mean, this is the whole thing; it's never really laid out formally

because they're treating methods that people were doing before analysis was really a thing

treating methods how? that second post is a fragment?

um, yeah, that was awkward word choice

:p

I mean, like separation of variables was just a thing people did for a long while before any good justification existed

integrating factor, similarly

and variation of parameters too

undetermined coefficients is at least honest about how it's playing fast and loose

fair enough

i just mean like most people consider differential equations with a heaviside forcing term to have a solution

but the solution is still continuous

but yet not differentiable

so the question arises: what is that kind of solution called?

piecewise differentiable?

:P

actually, that's fairly sensible

i mean... like how you said there are weak solutions

if you have reasonable pieces, a pw. differentiable function should have a continuous representative, just by translating the pieces together.

are there weak-strong solutions in between the smooth solutions and the discontinuous weak solutions?

Yeah, I mean, you could say 'has a solution' if you mean 'has a solution at all but finitely (or discretely) many points'.

hmm

that might open loopholes

perhaps I'll ask a question about it

seems like most people here have taken those lower level courses

like where you first touch laplace transforms and stuff

right

someone probably has a decent name for it

what /doesn't/ have a decent name for it?

many things :P

but they're hard to talk about jeje

@EricStucky regarding the piecewise differentiable shifting functions thing: well duh. You just summed up what I was doing in calc 3 and discussing originally about jump series nonsense.

and to be honest, it's probable most differential equations have been solved

^ no

wat

we can barely solve any ODE at all

certainly not analytically

the people over in #math on freenode irc were saying not to bother with trying to find more ways to solve differential equations. Distribution theory deals with all of them...

Or are they just crackpots?

trololol

no they're just a little misguided

ermmmm

wait

I know what they're saying

Perhaps we're talking past each other

not at all

They mean that distribution theory is the most general (reasonable) framework for formulating ODE.

to be honest, I thought they were wrong

Solving them, though, is a whole other can of worms

Here, actually I can do a bit better:

seemed like they were trying to say there's a (massive handwave) magic formula

if you can show that an ODE has no solutions in the sense of distributions, you can safely say that it has no solutions in any respectable sense.

i mean, if there were such a formula they'd just teach us the stuff to learn that

and not a bunch of other useless methods

(That's I think the most honest way of formulating it)

Hmm, I'm not sure I agree that they'd teach it to you, but you'd at least have heard about it

the way I figure it, the laplace transform is messy and so to avoid it we rely upon better methods to use at particular times

@EricStucky i meant like if it needed the gamma function they'd teach everyone the gamma function. They would sprinkle in little pieces leading up to it, and surely you would know of it!

@EricStucky to be fair though, those guys said I was a crank not for pursuing a weird solution but by "proposing a new system of calculus"

they are such idiots that they actually think I was trying to replace calculus. XD

I mean, I don't know when you were talking to them

You did sound kind of cranky for a while

about a week or two ago

i literally showed them my post on there and they claimed it was garbaage

bunch of trolls

i mean, I could see them saying it was garbage if there were already an efficient way to solve such differential equations.

but the closest I know of is the laplace transform

and that thing is not efficient

i certainly wouldn't program a computer to solve equations via laplace

hahahaha oh dear

me or them?

if you think laplace is not efficient, then you certainly would not like distribution theory >.<

I'm not saying the laplace transform is garbage

I'm not saying you're wrong

I'm just saying that I was very much misunderstanding what you were going for :P

well to be fair laplace transform cannot solve some of the things mine can solve

(granted the issue of continuous solution is certainly the downfall of my method)

i.e. finding it

well I'm also a programmer and probably more of that at heart

so while the mathematician in me likes the laplace transform as a tool to solve tricky problems, the programmer in me wants to make things a bit more efficient if at all possible

besides, engineers have to solve some of these problems. Always good to make their lives easier.

:p

jeje

fair enough

plus, it is just downright fun!

^

well I don't feel like asking a question right now

ill just add continuous to the weak solutions thing (granted the fact that I say continuous implied solutions should give that away)

I honestly think the weak solutions should be a new question, to be written later if you don't want to do it now :P

hmm?

it's just a minor edit.

technically it is what i meant when asking the question. I just didn't realize how things went, I guess. :p

didn't think it through

From what I understand: I don't think it is, though.

erm, sorry

frankenpost

From what I understand: the question with weak solutions is qualitatively different in some sense, because 'weak solution' doesn't have a consistent definition across all types of ODE

You're using a word that you don't really understand, which is fine, but does it make sense why this changes the nature of the question you're asking rather considerably?

fair enough

it's just im trying to ask about solutions in the sense that the course on differential equations was solving them

yet... I don't know what that is really.

:/

Honestly, I think that would be a great question

hmm

I'd be interested to hear someone who actually knows DEs

ill edit it back

give the answer

O.O

we didn't touch PDEs in our class

other than the notation anyways

haha sorry

slip of the fingers

:p

Yeah, I'm happy to do that if you're worried about writing a soft question b/c history

nah it's not that

just dont feel like writing it right now. XD

:)

feel ya bro

but by all means go ahead and ask it if ya want to

@EricStucky there is actually something interesting i might ask though as well

whether or not there is anyusefulness to the differential equation y'' + 2[x]y' + [x]^2 = 0