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12:02 AM
@VVKK77 Try this extension.
If you want to use it in mobile, take a look here
 
@leslietownes ChatJax doesn't work for you, either?
 
oh, sorry, it was a lame attempt at a joke. i meant that the latex i write doesn't render because i often leave the dollar sign markup out.
i'm either lazy or promoting an odd form of accessibility.
 
The perfect person is here if they do live up to their name

$ \sum_{i = 1}^{n}\bigg[\frac{X_{i} - \bar{X}}{\sum (X_{i} - \bar{X})^{2}} \bigg] = \frac{1}{\sum (X_{i} - \bar{X})^{2}}\sum(X_{i} - \bar{X})$

Are we allowed to "pull out" the denominator like that because the index $i$ is already been captured by a summation?.............that is....the denominator is just a number? @robjohn
 
12:17 AM
that's what it looks like to me, assuming that's what's intended. great example of why it's sometimes not great to omit an index entirely or use the same index twice.
that kind of stuff is all over my wife's stat books though. i get it, if you play by the rules everything begins looking like christoffel symbols, and nobody wants that.
 
yea in this case they both use the same index, but I didn't put it in the second one because it would look too clunky here.... I should state that though.
well I can't do that now...time has expired on edits :(
 
hello
What do you guys think of a new tag: "Catalan's contant"? I think these question have their own style involving very special and often hard series and integrals.
 
i have no opinion. i expect if i looked at the list of numbers that have their own tags i would agree with most but not all choices to include/exclude. assuming that other numbers have their own tags.
 
12:38 AM
$F:\Bbb R^n \to \Bbb R^n_+$ with $F(x,y)=(e^{x_1},e^{x_2}, \cdot\cdot\cdot,e^{x_n}).$ How do you show that $F$ is a linear map?
 
bruh
5
 
maybe just F(x) there? and remember some goofy operations on the codomain, not the usual ones.
 
yeah should be $F(x_1,x_2, \cdot\cdot\cdot, x_n)=$
 
@dc3rd Indeed. The $i$'s in the inner summation never see the $i$ in the outer summation. You could even change the $i$ in the inner summation to $j$ to make that clearer.
 
could someone glance at this deleted answer of mine math.stackexchange.com/a/4063668/27978 and, if you can guess, let me know why it might have been deleted?
not a biggie, but usually i have some clue, here i don't.
 
12:49 AM
Alright then. Thanks for the confirmation
 
@copper.hat There are some people here who think that hints are bad, and that only complete answers are good. Often, these same people think that giving a complete answer to a question with little effort is bad. However, two answers were provided with lots of work to them, one claiming to be a hint. Those were only posted recently, as was most of the work in your answer. With the improvements to your answer, I think it can be undeleted.
 
Any thoughts on my question?
 
@copper.hat I have undeleted the answer.
@geocalc33 why do you think it is linear?
what are the requirements of a linear map?
can you show they hold or don't hold?
 
1:05 AM
the linear structure on the vector space $\Bbb R^n$ given by: $x_1+x_2 \cdot\cdot\cdot x_n$ transports to $x_1x_2 \cdot\cdot\cdot x_n$ in $\Bbb R^2_+.$ Also scalar multiplication $\alpha x_n$ is analogously given in $\Bbb R^n_+$ by ${x_n}^{\alpha}.$
@robjohn
 
Is that really how they are using $\mathbb{R}_+^n$?
 
I think $\Bbb R^n_+$ is the image of $\Bbb R^n$ with the usual vector space structure under $F$
 
If that is so, then that needs to be stated ahead of time since that is not something I have seen as standard.
 
So I think that $F$ is a linear map because it preserves linear structures
 
goofy operations. i found it helpful in contexts like this to invent funny symbols for the goofy operations.
but some people hate that.
 
1:10 AM
@geocalc33 what you're searching for is group isomorphisms
 
@geocalc33 so show it obeys the rules that a linear map needs to obey.
 
although I'm not sure that's what you mean in n coordinates
 
for example if scalar multiplication and 'funny' and not what is usually meant by the juxtaposition of two symbols from your set, maybe invent a funny symbol for it.
 
@geocalc33 what happened to the $y$ in the definition of $F(x,y)$?
 
it realized it wasn't wanted.
 
1:12 AM
@leslietownes poor "$y$"
 
whatever. y's been kicked out of better problems than this.
 
@geocalc33 I mean, if you just consider the coordinatewise evaluation you had above it is still a group iso, but I don't know why you're now multiplying all coordinates
 
he's regarding it as a vector space, not just a group. hence not just a goofy addition law but a goofy scalar multiplication.
 
yeah I'm telling him that vector spaces are not appropriate here
 
@dc3rd You're not allowed to use the same subscript.
 
1:14 AM
they might be. i agree that $\mathbb{R}_{+}^n$ should probably have been defined somewhere as to make the operations clear. before you even ask if something into it is a linear map. but we live in an imperfect world.
 
But they're summing over the same "items"
 
The letter doesn't change the items. But you can't abuse poor $i$.
I would never write $x_i/\sum x_i$
$i$ is both fixed and varying. You get it?
 
@user2103480 okay I may have written something incorrect above. I can write it well in one dimension, like $F:\Bbb R \to \Bbb R^+$
 
This is subtler than I thought it would be..............so even though I am summing over the same items in the same order I can't use the same index. So to distinguish between possibly different items it revloves around using different variable names. So if I had a situation with $X_{i}$ and $Y_{i}$ for instance
Yea I get what you're saying there Ted
 
Not subtle at all
 
1:19 AM
$a+b$ in $\Bbb R$ goes to $ab$ in $\Bbb R^+$ and $\alpha b$ in $\Bbb R$ goes to $b^{\alpha}$ in $\Bbb R^+.$
 
Your second point is not the issue. You can put $i$ on both X and Y if it has the same meaning, of course.
 
the important thing is if one is fixed, then it remains fixed, but easier said don't abuse poor $i$ and let him feel unique and special.
 
There you go. No Karen $i$.
 
when does the tensor product of nonzero R modules are nonzero? where R is a commutative ring with unity
product of two nonzero modules
 
Answer yourself, @love_sodam
 
1:21 AM
lol
 
some good questions on math.SE on that issue.
 
Democracy crashing and burning today in the US.
 
ok let me search that
 
by good, i mean responsive. i don't mean subjectively good to me. i don't want to tensor commutative rings together and you can't make me.
 
tensors with a weapon at leslie's back
 
1:24 AM
it's Democracy 2. now with fewer voters.
 
@robjohn Much appreciated!!! I am always curious about downvote reasoning, and delete votes amplify that curiosity.
 
Not 2.0 ... after that
 
commutative algebra would be a lot easier if we had a universal answer as to when the tensor product of two rings is zero
 
when viewing $\Bbb R$ and $\Bbb R^+$ as different vector spaces, the $\exp$ function is linear. But how do you prove that?
 
@robjohn My original answer had a typo. (a missing $2$) and was very definitely of the 'hint' variety.
 
1:25 AM
geo, you verify the axioms, if indeed they hold. same way you prove any explicitly given simple enough map with a formula is linear.
 
@dc3rd The $i$ in the inner or denominator sum is bound to that sum. It's scope is limited to that sum. The $i$ from the outer sum does not extend inside the inner sum. The problem with overloading the veriable $i$ like that is that the $i$ from the outer sum cannot be used inside the inner sum.
Ted and I evidently have differing views here
 
i do wonder if it might help not to use juxtaposition for any operation in $\mathbb{R}_{+}^n$. call the addition operation by some funny symbol like $\oplus$ and the scalar multiple operation by $\circ$. it'll look more linear when you do that, trust me. or if it doesn't it will point the way to why it isn't linear.
 
We do?
Oh, we do. You will confuse students for sure with your rules.
 
@TedShifrin Well, I think that the $i$ can be used as it is in dc3rd's statement, but you said it is not allowed
 
is a commutativity a very very special property?
 
1:28 AM
i had a lot of friends who had that view. if you bury something small inside a big expression and you only sum in that small thing, the dummy variable is eaten and can be reused. i think it's confusing.
 
Yes, my 40 years of teaching lead me to be unambiguous.
 
@leslietownes okay
 
i mean in specific instances i can resolve it, but when i have written sums it's usually because i want to manipulate them symbolically, and i can't have second-order notational conventions in my head while i do that. i want to just compute.
 
Damn, leslie and I agree again.
 
we live in strange times
 
1:29 AM
@robjohn You seriously will write $x_i/\sum x_i$?
 
i do that often
 
The hell with you.
 
@TedShifrin I don't use this in ambiguous situations, but it is used that way and so I try to explain how scoping and binding works in case they encounter it.
 
Just f******* avoid it.
I will go back to burying democracy.
 
@TedShifrin I would not use it without a binding variable in the sum, but I try to avoid it.
 
1:31 AM
It was a notation from a text I was reading and while I was trying to understand the steps the author took to arrive at the solution, the situation we're discussing came up
 
I have seen $\frac{x_i}{\frac1n\sum\limits_{i=1}^nx_i}$
Though I would avoid it myself
 
he didn't use any indices at all, but I assumed he meant he was attaching $i$ on both so to get clarification I asked
 
And if the author had listened to me, you would not have been confused.
 
but in practice it makes a lot more sense myself to avoid it when working
 
Define $\bar x$ and be done with it.
 
1:32 AM
think of all the confusion, added up over time, from that. it's like greenhouse gas emissions.
 
@user2103480 why aren't vector spaces appropriate here? If $F$ is linear when regarding $R$ as a group then can't you do the same thing regarding R as a vector space?
 
@dc3rd yes, it does present a huge risk of getting things mixed up
 
maybe we can put disambiguable-but-kinda-ambiguous notation on some kind of blockchain and burn fossil fuels to create more of it.
 
LOL
 
geo-coin
 
1:33 AM
sometimes you need to "massage" $\bar{x}$ into something else @TedShifrin....you know? symbol shifting.....:)
 
geocalc-coin
 
Nah.
 
geocalc33-coin
Bit-coin
 
i'll run a trademark clearance search.
 
Jesus-coin
 
1:37 AM
@dc3rd yeah, that is the $x$ that lives on the other side of the real axis.
I've always wondered why the definition of variance involved complex conjugation.
 
Blah.
 
$\frac{\overline{\Xi}}{\Xi}$
 
@Thorgott 60 minutes ago you said "bruh." Did you say "bruh" because I left out important information, used incorrect notation, or asked about this map many times before?
 
Is all of the above an option?
 
the importance of context................I haven't directly encountered it, but seen it in passing, but doesn't some stats use complex numbers in some way? So I could even see $\bar{x}$ causing issues for somebody not in the know.
 
1:43 AM
@leslietownes the direction of Xi?
 
now that's what i call unambiguous notation.
 
You do?
 
@geocalc33 he sometimes just shouts bruh
don't worry about it
 
Oh, worrying about it is recommended.
 
@geocalc33 ok so a thought experiment
you have a vector space where every vector is of the form $(e^{x_1},...,e^{x_n})$
 
1:56 AM
Doesn’t sound like much of a vector space. Hmm.
 
EM4
@TedShifrin thank you for the integral domain, I totally get it. Sorry for the late reply :).
 
That was years ago!
 
EM4
yeah I know. I got super busy and was like I never said thank you.
 
No biggie.
 
it is good to express gratitude.
 
1:58 AM
@TedShifrin the problem is how does he define multiplication etc.
by scalars
 
I have tortured EM4 plenty :)
 
EM4
this true, one of the best people ever.
it got me thinking deeply :)
 
t * (a,b,c) = (a^t, b^t, c^t) might do nicely
 
EM4
I was doing my homework I was like yes!
 
What is the zero vector?
 
2:00 AM
1
 
and if he just defines addition and multiplication via the isomorphism to R, one actually gets a vector space, but this is a tautology
 
don't think too deeply. it's worse than not thinking at all.
 
EM4
@TedShifrin do you know any books for graph theory.
 
no, not my expertise.
 
I too thought he meant literally adding the numbers, but yeah this is a vector space since it's just R^n
 
2:02 AM
"just R^n." what if R^n heard you say that.
 
so my next question is what does $\text{SL}(2,\Bbb R)$ look like in the new but "same" setting
in $\Bbb R^{2+}$ as opposed to $\Bbb R^2$
 
2:24 AM
So what if we had $\Bbb C$ (or maybe the affine plane, I am not sure which is better) and had concentric annuli about the origin (so probably not the latter because we have some origin) such that you have one of "thickness" 1/2 with outer edge at 1 and then one of thickness 1 with outer edge at 2, and so on, and then the same within, but instead of doubling the thickness you halve each time. So now identify all the annuli. What space is this?
 
2:45 AM
i always think of the affine place as something found flying around the bermuda triangle with no known origin.
 
3:11 AM
@copper affine place or plane?
 
uugrh, i always misspell a crucial word!!!
 
What would be perpendicular to X , Y and Z axis ?
Can we say nothing or there can be sth ?
 
3:55 AM
Whenever I am taking average particularly the weighted average I often get confused about what I am supposed to take in the denominator (whether that is the weight or number of variable) and most importantly why?
Weighted average seems to be too difficult for me..
A slightly detailed insight into the same is most certainly welcome.
It would be nice extremely kind of you if you could assist me with this..
 
4:07 AM
Hello! @robjohn can I ask something?
 
As it says in the sidebar: Just ask; don't ask to ask. In other words, yes.
 
Oh, sorry. I tried using your bookmarklets on Google Chrome, Android, but it seems like it doesn't work. Is there a w
Is there something that I didn't do?
 
@RajorshiKoyal As with ordinary averages, you are looking for some $\bar{x}$ such that $\sum_k w_k x_k = \sum_k w_k \bar{x}$. so $\bar{x} = { \sum_k w_k x_k \over \sum_k w_k}$.
 
does my question not make sense or is it not interesting maybe?
 
I think he wants a full explanation on what a weighted average is@copper.hat, in which case I refer him to the numerous sources a google search can bring up.
 
4:12 AM
@soupless I'm not sure what the problem is. Some people have trouble with Chrome.
4 hours ago, by Wolgwang
@VVKK77 Try this extension.
 
I do like your simple way of expressing the problem though.
 
@dc3rd Maybe he/she might have some physics/engineering background with center of masses of non uniform objects?
 
Not from the questions they've been asking over the last couple of days they don't.....
 
must be exam time...
tough love is best but i am a sympathetic wimp when someone needs help
 
@robjohn It does not work, unfortunately. I'll try finding another way, thanks!
 
4:18 AM
Lol....the student side of me empathizes with this, the idealist in me disagrees...
 
@soupless So my bookmarklet doesn't work and the chrome extension doesn't work. I wonder if there is some browser setting that needs to be tweaked.
 
@robjohn It works on Samsung Internet, but not on Google Chrome. Let me first try on Firefox.
 
@soupless which works where?
or do both work there?
 
@robjohn Your bookmarklets work on Samsung Internet Browser (I hope the name is correct)
 
@soupless what OS are you running?
 
4:28 AM
@robjohn Android.
 
Ah, yes, you mentioned that. Ok, just trying to keep track of where the problem appears.
 
@copper.hat One of the many different exams in India ...
 
@robjohn I think it will or does not work on Firefox, as stated here: support.mozilla.org/en-US/questions/1301329
 
I'm trying to discourage his coming here, posting problems repeatedly, and demanding we help him. His tone has not changed much.
I will start kicking him, but when people here keep helping, I can't do that.
 
@copper.hat Thanks a lot for replying.
 
4:38 AM
@soupless I use Firefox, and Chrome is the only source of difficulties so far.
 
@robjohn Ok, maybe I still don't know how, but whenever I tap the bookmark with the script, it becomes a search query. How do you make it so that it will not be like that?
 
@robjohn I use it constantly on Chrome and Safari.
I quit using Firefox except on rare occasions. Now I don't remember why.
 
@TedShifrin Ah, great! I had not heard whether anyone had used it successfully on Chrome.
 
Oh, I have used it on my desktop with Chrome for years.
 
So it is not something that the bookmark does that Chrome does not support. That is good news.
 
4:44 AM
I even have it working on my phone/iPad with Chrome and Safari.
Now these are on Apple products. Who knows about Android.
 
@TedShifrin I use it on my iPhone running Safari
Though I hate using chat from my phone
 
I use your applet on both on both Safari and Chrome. Yes, anything with MSE sucks on a small device.
 
The mobile interface is just not as friendly to me
 
The big problem for me is typing all the ChatJax ... it's just too tedious and I make zillions of errors.
or MathJax on main on the app.
 
@TedShifrin I have a small scratch pad that is an html text file with the MathJax header to it. I use that and BBEdit will render the HTML, MathJax and all.
 
4:48 AM
Oh, that sounds intriguing.
But, still, for me typing on iPad/iPhone is an ordeal.
 
That way I preflight any long mathjax and I can even save it for later.
@TedShifrin It is not as nice as a standard keyboard
 
I can use a separate keyboard bluetooth with my iPad, but I hardly ever bother.
 
Yeah, using a keyboard with an iPad seems cross purposes
just use the laptop
I don't even have an iPad. I have an iPhone and a laptop. In between just seems excessive.
oh, oh, I need something with 3/4" less screen size today...
 
I don't have a laptop. Sometimes I am on MSE when I'm not at my computer or even not at home (like with the pandemic)..
 
typing out MathJax on the app on android is a nightmare....most of the time it doesn't render.....
Now I just acknowledge that "that isn't going to be a math time for me" if I'm left to only using the app from a phone.
 
5:01 AM
Do you have robjohn's applet on your android? It does render fine on my phone and iPad. But I agree that it is too tedious.
So ... any pithy math question, @dc3rd?
 
The StackExchanges haven't made the chat accessible on ANdroid, I read it was in their roadmap, but this was documented in 2016.....and well we are here....
 
Oh, interesting. I didn't know that. But glad to know that we Apple folks are once again superior.
 
eye roll...................
 
Careful. If you roll too many eyes at me, I'll just ignore you in perpetuity.
 
At least you're an Apple user that acknowledges the existence of the divide.....too many times I'm greeted with a kumbaya from Apple users saying the divide doesn't exist.....lol
 
5:06 AM
or someone will compute the probability of rolling 6 eyes in a row.
 
Well, I suffered through teaching a calculus lab many years ago on Windows machines and it just made me like Apple more.
@robjohn: But I usually roll a transcendental number of eyes in here. Some people actually keep track.
 
used to be an apple fan in the apple 2 and early mac days. not a fan of jobs definite fan of woz.
 
I've alsways been under the impression/idea from things I've read that Apple computers can't do the "grunt" work of difficult math..
maybe I'm just having an attributive bias because of my android and Windows habits
 
Do mathematicians think of $\pi$ or $e$ during transcendental meditation?
 
the rest of my fam are apple fan people
algebraic meditation?
 
5:09 AM
When I need to be discrete about it, I only do integral meditation
 
sumthing to think about
 
@dc3rd: I think in the early days applied math people thought you couldn't do serious programming on Macs, but I think those days are gone.
@robjohn better to be discreet.
 
There you go spilling the spelling on my jokes.
 
I thought you intended discrete meditation.
 
I did, but I think the statement was in a superposition of states.
 
5:12 AM
Apparently so. No apparitions yet, though.
 
crying statues?
 
hmmmm........I don't know if I can fully trust you about this Ted....you are part of the cult..............you won't Jim Jones me.....no sir.
 
@TedShifrin Are you intimating that the reals are supernatural?
 
are you a flat manifolder?
 
Where did the reals come from here?
I'm fine with your not trusting me, @dc3rd.
 
5:14 AM
After God made the integers.
 
that is the showing of a man with cast-iron self assurance.......I like it.
 
Some would say arrogance.
@robjohn The stoopid bot is on the loose again, activating posts like this from zillions of years ago.
It really is quite tiresome.
 
@TedShifrin that's the 4th time in the last year that has been bumped.
 
Why? Almost everything I click on now is from 2016 or 2017. Why is this happening?
 
sounds like the sort of thing someone seeking assurance would say.....but now it starts to get too philosophically deep.....back to working with R I go....bye folks.
 
5:23 AM
Ding ding.
 
Bye, @dc3rd.
 
@dc3rd that's right, just bye folkal us
perhaps I just need more sleep.
 
LOL ... um ...
 
oh, no. I just realized that I answered a diff geo question tonight. Hush!
 
That was hardly differential geometry. Tags are usually abused.
I just don't have the patience/energy to change all the wrong tags.
 
5:35 AM
oh, it had $\mathbb{R}^n$ and orientation and all sorts of differential geometry words in it.
 
I don't see connections or curvature.
Therefore, no differential geometry. Just a multivariable analysis/calculus exercise.
 
wow... a tough audience
 
@TedShifrin what's happening in the US?
Also, hi all
 
Lots of states, led by Georgia (whose governor rushed things through and signed the bill the same day it was passed), disenfranchising voters left and right. If you can't win legitimately, lie a lot and make it impossible for your opposition to vote.
 
Damn :/
 
5:40 AM
My post on FB was "bye-bye democracy."
 
Yeah the collapse of democracy is happening in the UK too
the English speaking world is a joke lol
 
hey @TedShifrin does this make sense as a question:
https://chat.stackexchange.com/transcript/message/57448464#57448464
 
Yes, UK still has Trompie Boris, and we still have virtually every Republican (white) senator/representative on the side of white supremacy.
 
Have you heard that the Conservatives under Bojo are trying to get a bill passed that restricts the right to peaceful protest? And they've also introduced a rule that government buildings must fly the Union Flag every day of the week
 
@BigSocks, so you're filling up $\Bbb C-\{0\}$ with these annuli? Why do the actual radii/thicknesses matter?
@Edward: It is appalling how quickly we've gone back a century.
 
5:45 AM
Yup :(
 
well the question is mostly motivated by this one videogame I saw a friend play (Maquette) where the action happens in a space like this
the thickness matters because the space has a property that is kind of like this
and I wondered what space this actually was
 
Not mathematically. So you're identifying just the edges, or the interiors of the annuli too?
 
the interiors too
 
So, it's just turning $\Bbb C-\{0\}$ into one single annulus.
 
there are items that you can carry "between annuli" and then you can move "back and forth" between annuli that were before coinciding
 
5:47 AM
Pick your favorite one. All the points everywhere else are equivalent to a point in that.
 
kind of yes
 
Maybe the video game isn't quite what you've proposed. I dunno.
 
@EdwardEvans so now we can only have unpeaceful protest?
 
@robjohn Lol well that's what's been happening over the last week or so. The bill says that any protest that causes "public nuisance" is illegal and protesters can be given up to 10 years in prison for participating.
 
the thing is that if you take something from an annulus $i +1$ to an annulus $i$ the item becomes larger in the $i+1$ version, so maybe the identification is not accurate
@TedShifrin yeah, I think this is true
 
5:49 AM
@EdwardEvans I hope that falls as flat as it should.
 
it is very strange because of this "enlarging items" property
anyway, at least there is not obvious analogue in math for such a space
 
Topology doesn't see sizes. Your equivalence relation is not an isometry, of course, but that doesn't affect my answer to your question.
There is not a metric on one annulus that induces corresponding metrics on the equivalent annuli.
Oh, actually, there is. It just automatically magnifies as you blow up the radius. That's natural enough.
 
@robjohn Unfortunately the UK is chock-full of gristle heads who have been convinced that the bill is necessary. It doesn't help that the leader of the opposition is absolutely useless (and is being described as a tory plant and called "The Red Tory").
 
right I imagine really moving "between annuli" is just picking different elements of the same equivalence class, but all of them "happen at once"
which I think reflects the "automatically magnifies as you blow up the radius" aspect of what you said
 
I suspect this isn't so interesting mathematically.
 
5:53 AM
yeah, this is also my suspicion
but on the off chance it was cool I figured I might ask
thanks for considering it anyway :)
 
What does show up classically in video games is that a lot of video games are actually on a torus. Left and right edges are identified, and top and bottom edges are identified. If you flipped one of them, your video game would be on a Klein bottle.
 
right and that is pretty cool too. but this is definitely something I had never seen. It is called "Maquette" for what it's worth
 
6:06 AM
@TedShifrin that's interesting
 
 
2 hours later…
8:20 AM
Hi, if $X$ is a set for which there is a filter $\mathcal{F}(x)$ of sets containing $x \in X$ assigned to every point $x \in X$, and these filters are such that for all $U \in \mathcal{F}(x)$ there is some set $V \in \mathcal{F}(x)$ such that for all $y \in V$, $U \in \mathcal{F}(y)$, if we say a set $O$ is open if it is empty or such that for all $o \in O,O \in \mathcal{F}(o)$,
then why is it true that the closure of a set $A$ is the set of all points $x$ for which all members of $\mathcal{F}(x)$ have nonempty intersection with $A$?
I can prove this if I know that every member of $\mathcal{F}(x)$ contains an open set containing $x$, because then it just comes down to proving the same statement with 'neighbourhood' replaced by 'open neighbourhood' , but in its current form I don't know how to relate members of $\mathcal{F}(x)$ with open sets containing $x$
 
 
4 hours later…
12:19 PM
hi all
i have 2 questions about fourier series
0
Q: $\exp(a_0 + a_1 \sin(x) + a_2 \sin(2x) + a_3 \sin(3x) + ...) = b_0 + b_1 \sin(x) + b_2 \sin(2x) + b_3 \sin(3x) + ...$?

mickI was thinking about fourier series. Alot is known about it. But I wonder $$\exp(a_0 + a_1 \sin(x) + a_2 \sin(2x) + a_3 \sin(3x) + ...) = b_0 + b_1 \sin(x) + b_2 \sin(2x) + b_3 \sin(3x) + ...$$ Is it possible to "simply" express $a_i$ in terms of $b_i$ or vice versa ? I use "simply" because ofcou...

0
Q: $ \int_0^{2 \pi} f(x) \sin(nx) dx = n^{-4} \int_0^{ 2 \pi} \frac{d f(x)}{d^4 x} \sin(nx) dx $?

mickI was thinking about fourier series. So I wondered : Can we describe all solutions $f(x) = - f(-x) $ such that $$ \int_0^{2 \pi} f(x) \sin(nx) dx = n^{-4} \int_0^{ 2 \pi} \frac{d f(x)}{d^4 x} \sin(nx) dx $$ for all integer $n>0$ ?

 
 
2 hours later…
jay
1:58 PM
What do people call the limiting solution of a PDE?
Like as time tends to infinity say the solution of the PDE converges to something, what do you call that ? the limiting distribution ? limiting solution?
 
2:38 PM
@Wolgwang Thanks for the information. That's really interesting.
 
2:56 PM
@jay What do you mean by "the solution converges to something"?
 
3:27 PM
If P_1,...,P_n are distinct prime ideals in a commutative ring R that don't contain each other, then what are maximal ideals of $(\bigcap_{i}(R-P_i))^{-1}R$?
 
Can someone help me wit this? Thanks!
0
Q: KKT conditions for quadratic optimization

statwomanAssume the primal problem of: $$max_x \mu^Tx$$ $$st. x^T\Sigma x\leq \sigma^2$$ $$ Ax=b$$$$ Cx\geq d$$ What is the KKT optimality conditions for this? So far this is what I got: $$L(x,\lambda,w,v)= -\mu^Tx+\lambda^T(x^{T}\Sigma x-\sigma^2))-w^T(cx-d)+v(b-Ax)$$ $$x^{*T}\Sigma x^*\leq\sigma^2$$ $$-...

 
3:46 PM
The Wikipedia article on Fourier Series says, "Through Fourier's research the fact was established that an arbitrary (at first, continuous and later generalized to any piecewise-smooth) function can be represented by a trigonometric series." How is it the case that piecewise-smooth functions are a generalization of continuous functions?
Should this perhaps say "extended to also include any piecewise-smooth" instead? Is a function's being either continuous or piecewise-smooth the necessary and sufficient condition for it being represented by a Fourier Series?
 
i wouldn't trust anything on wikipedia relating to fourier series. it's a little too vague in an area where precision really matters.
what do they mean when they say 'represented by' a fourier series, for example. you can compute fourier coefficients (and hence a series) for much larger classes of functions than piecewise smooth functions. they don't always converge to the original function pointwise, or at least not everywhere, but there are still an array of senses in which such functions may be said to be 'reprented by' their fourier series.
 
Gotcha, yeah based on what you've just said, it sounds like the questions about Fourier Series in my head are a bit too simplistically framed.
 
i used to have a translation of fourier's original paper. it had some of the main idea but (unsurprisingly) was light on what we would consider rigor. i think he understood the case of continuous functions with left and right hand derivatives at every point. i don't recall that he did anything without some hypotheses resembling differentiability.
a good resource for the classical treatment of this is tom koerner's "fourier analysis." a lot of more modern books focus too much on modern theory, which is less focused on pointwise convergence because pointwise convergence is really difficult to prove for a lot of functions, even when it happens.
for example the fourier series of an L^2 function converges pointwise almost everywhere to that function. this was suspected for about as long as people knew how to phrase "L^2" and "almost everywhere" (whether precisely or imprecisely) but not proved until 1966, which not knowing your age i will add was not that long ago, mathematically speaking. the guy who proved it is still alive.
and the proof is impossible to read. there are shorter proofs with more modern techniques
 
@leslietownes what is your focus in math? functional analysis?
 
i don't think it's unfair to classify a lot of the study of fourier series, from fourier to the mid 20th century, as focusing way too much on something that is too difficult [pointwise convergence] and then giving up and discovering other more useful things because people gave up.
my focus was functional analysis. i am no longer active
i take back some of what i said about wikipedia. en.wikipedia.org/wiki/Carleson%27s_theorem is pretty good.
that's a common feature of wikipedia, i think. the broadest topics have the most vague junk in them, not because expository choices are being made, but because a lot of the editors just don't know enough about the subject to put in the two or three more words that would make it make sense.
then you deep dive into something and find pages on individual theorems that are pretty good
en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors is a real circus, for example. just a spew of facts about the subject with little organization or indication of what might be more fundamental and what might be an aside. it can't decide if it's trying to tell me about eigenvalues, or be an entire linear algebra book.
 
4:02 PM
lol
I found a resource at my level finally - it's refreshing to read
really well written
 
Thanks for the insight! I've only been exposed to Fourier Series in an ODE, and currently in a PDE course; I haven't studied it as a subject of its own ($L^2$ is a new term to me as well). My curiosities about Fourier Series are definitely on the less rigorous side, i.e., questions like "Would it be possible to represent all functions as infinite linear combinations of square waves, triangle waves, or sawtooth waves instead of sinusoid waves?" and "What sorts of functions can be represented
with one wave instead of two," but it sounds like these questions are too vague without specifying what is meant by representing a function with a series in very precise detail.
 
often when people say 'represented' they mean 'be the pointwise limit of, everywhere' but this breaks down very quickly even with saw waves or any functions with jump discontinuities. and they knew that very early on, so already you have to relax a bit.
 
This can be solved by weighted sum isn't it?
If yes then how do I solve it...???
I need to add the weights in the denominator right..so that should be lets say sum of the precentages..
This is particularly from an exam that demands faster calculation.
 
and once you get used to a fourier series's pointwise values not matching your function at finitely many points, you then wonder, well, what if i'm OK with not matching at countably many points, or what if i'm fine with it as long as its integrals over intervals are the same. etc.
 
So necessarily I am asking how do I arrive at the solution faster in a sense that I take a single step average ...Can I get a detailed intuition into the technique..A guidance in the direction is most certainly welcome....
I mean if some extra information is required then please tell me...
 
4:18 PM
@leslietownes Awesome, that helps to clarify how I should be thinking about these types of questions. Thanks again for the explanation :)
 
4:34 PM
@leslietownes Would I be asking for too much if you could bear with me for sometime..
 
@RajorshiKoyal I've asked you politely four times now. Please find other people at another site to pester with your repetitive questions. I will kick you out of the room.
 
@TedShifrin Please do not kick me out..I am sorry.
 
Go find friends to study with. We are not here as exam preparation for elementary mathematics and statistics.
 
ok..sure..
 
I mean it.
 
4:52 PM
I found a loop hole in MATLAB licensing ...should I report it
 
only if you invoice them for your consulting services and the check clears first.
 
Are you a professional finder of loopholes? Didn't you find a loophole in something else recently?
 
i remember meeting john little at a conference many decades ago and thinking "there's no money in selling matlab".
 
he's going to break the world record for most software licenses actually read start to finish. i think it currently stands at 3.
 
@TedShifrin JEE main exam login
 
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