12:26 AM
do you guys have a visualization of the triangle inequality you use?
I keep confusing the order of the triangle inequality and the reverse triangle inequality

It’s all triangles, silly!
The longest and shortest the third side of a triangle can be. (Think vectors, not real numbers.)

oh
should've thought of that hehe
thanks!

Yup.

1:31 AM
visualization? no thanks.

2:01 AM
smacks @Leslie once and for all

Prove $-|f(x)| \leq f(x) \leq |f(x)|$.
Either $f = -|f(x)|$ or $f = |f(x)|$, and:
$-|f(x)|, |f(x)| \in [-|f(x)|, |f(x)|]$
$\therefore f \in [-|f(x)|, |f(x)|]$
$\therefore -|f(x)| \leq f(x) \leq |f(x)|$
faster way?
found it:
Either $f = |f|$ or $f = -|f|$, and $-|f| \leq |f|$.
$\therefore$ $-|f| \leq f$ or $f \leq |f|$
$\therefore$ $-|f| \leq f \leq |f|$

2:33 AM
you like extraneous symbols :-). just $-|x| \le x \le |x|$ would be sufficient to prove...

hm, wouldn't that one depend on what $f$ does to its inputs?

huh?

we get $f(-|x|) \leq f(x) \leq f(|x|)$

if the above is true for the symbol $x$ it is certainly true for $f(x)$.
good god. what are you doing?

ever greater speeds of analysis

2:40 AM
i think you need to slow down, you have skidded off the road :-)

but, more exactly, the lipschitz diff eq existence theorem

your third last comment is wrong unless there is something about $f$ that I do not know.

that's why $-|x| \leq x \leq |x|$ bothered me

if $-|\text{symbol}| \le \text{symbol} \le |\text{symbol} |$ you can replace $\text{symbol}$ by whatever you want and it will still be true.
i hate to use the term, referential transparency applies.

it's even true for symbols that repreresent numbers.

2:54 AM
reprehensible and resentful ones, that is?

they're reprehensible and i'm resentful

3:38 AM
@shintuku Huh?

symbols repreresent numbers, as it is known

4:00 AM
@schn little-o is really only a one variable family. When we have two variables involved the estimates may or may not be uniform in the second variable. These need to be handled on a case-by-case basis, which is why the calculus of little-o that you seem to want may not always be what you want.

4:15 AM
woah! you can make nested absolute values dissapear by using the property that you can always add a negative within the absolute value brackets
useful for nested absolute value integrals

4:36 AM
does rudin have something on the existence theorem for first order differential equations that states that approximations to the solution of an initial value problem converge to the solution?

5:20 AM
@shintuku shin, did you refer to the exercise problem I suggested few days back?

5:57 AM
Though that exercise problem doesn’t talk about approximations.
but the exercise does give a sufficient condition in which initial value problem will have at most one solution.
I’m referring to Exercise no. 27 in chapter 5 of Rudin’s PMA

6:16 AM
@Koro I'm actually working towards that exercise
it was too hard back then but I'll be able to soon

2 hours later…
7:58 AM
@shintuku :-)
I am given: a recursively defined sequence: $a_1=0, a_2=1$ and $a_{n+1}-2a_n+a_{n-1}=2$ for $n\ge 2$. I want to find nth term and lim $a_n$.
It is easy to see that the limit won’t exist finitely because if it did then recursion relation is violated.
I am looking for a way to find nth term.
Writing $a_{n+1}$ in terms of indices n-2 and n-3 and so on, seems complicated. Is there any other way?

8:36 AM
@shintuku what?! that doesn't sound right. Can you give an example?

what statement?
the nested integral absolute values?

@shintuku the one I referenced.
4 hours ago, by shintuku
woah! you can make nested absolute values dissapear by using the property that you can always add a negative within the absolute value brackets

oh first time I notice the reference feature

really?

yeah hehe
here's an example:
$|\int_{x_0}^x |t-x_0| \ dt| = |-\int_{x_0}^x |t-x_0| \ dt| = | \int_{x_0}^x -|t-x_0| \ dt|$
$\therefore |\int_{x_0}^x |t-x_0| \ dt| = |\int_{x_0}^x t-x_0 \ dt|$
right?

8:42 AM
The last is true only because $t-x_0$ does not change sign on the domain of integration.
$\left|\int_1^3|x-2|\,\mathrm{d}x\right| \ne\left|\int_1^3(x-2)\,\mathrm{d}x\right|$

It could for negative $x$ no?

what do you mean could? you are stating something as a rule.

professor @rob, for this question of mine:
43 mins ago, by Koro
Writing $a_{n+1}$ in terms of indices n-2 and n-3 and so on, seems complicated. Is there any other way?

it seems like the way which we use for solving homogeneous linear ordinary differential equations with constant coefficients.
I had never seen it before. I wanted to understand how to choose the particular/trial solution in case the recursion is not homogeneous. My understanding is that it is chosen by hit and trial

8:50 AM
@Koro not differential equations, but difference equations. There is a difference.

For example if I have a recursion: $x_n= a x_{n-1}+b_{n-2}+\sin n$, then what can we choose as particular trial solution. In case of differential equation, if I remember correctly we choose A cos n +B sin n. But here?

argh I thought I had a new rule to deal with that sort of expression faster
thanks for pointing that out

@robjohn I meant that the method on that link looks similar to the one used in case of linear ordinary homogeneous differential equations such as $y"+ay'+b=0$ where we assume the solution to be $y=e^{\lambda x}$ and solve for $\lambda$.

I was using the fact $|f|=-|f|$ implies that $|f| = f$. Don't we have that in the above case? (I realize you're right but I'm trying to figure out where I went wrong)

$|f|\ne-|f|$ unless $f=0$

8:56 AM
hmm right :)

oh! you're right!

@Koro No. That gives the wrong answer for a difference equation

So how can we choose trial function for "difference equation" professor Rob?
For example here:
6 mins ago, by Koro
For example if I have a recursion: $x_n= a x_{n-1}+b_{n-2}+\sin n$, then what can we choose as particular trial solution. In case of differential equation, if I remember correctly we choose A cos n +B sin n. But here?
Though, I have never faced such question before so my guess is $A\cos n+B\sin n$

@Koro You need to learn about difference equations before you start tackling something like that equation

Ahh, okay professor @Rob

8:59 AM
I am trying to find a good reference

yeah that will be very helpful professor @Rob, thank you so much :)

is it also in general the case that $|f| = |-f|$?

yes @shin

hm

@Koro I remember this Wikipedia article being clearer, but take a look at it.
@shintuku that is by definition

9:02 AM
hm.. I inferred $|t-x_0| = -|t-x_0|$ from $|\int_{x_0}^x |t-x_0| \ dt| = |\int_{x_0}^x -|t-x_0| \ dt|$
is that not the case?

@robjohn thanks a lot professor Rob. I'll study that. :)
@shintuku the inference is wrong

ah there you go, that's where I went wrong
I'll figure out why, thanks for all the help

:)

@shintuku the equation on the right is true, but the equation on the left is not

professor @Rob, are you awake too early today?

9:10 AM
@shintuku The equation on the right is true because $|x|=|{-x}|$ not because $|x|=-|x|$

right, that makes sense
I thought I could do the inverse operation on the absolute value and integral on both sides, but I'm realizing I didn't think of the inverse operation of $\int_{x_0}^x$

@Koro once you can solve that equation, you can start to look at the inhomogeneous equation

alright professor Rob, thank you so much :)

oh, nevermind the integration: $|f| = |g|$ does not in general imply $f = g$
that's what I get for using the $\sqrt{f^2}$ form of absolute values
second time in two days I've been saved terrible headaches in the future by mathse chat
bless you all

10:02 AM
@robjohn By “…uniform…”, do you mean that it is not clear what happens to $u$ when $h\to 0$?

@schn well, I mean that for different $u$, the convergence of $\frac{f(h)}{h^m}$ to $0$ might be altered. The limit may be $0$, but the rate of convergence might change. It depends on the context how and if this affects the result. That is why it is difficult or impossible to create a calculus for little-o.
basing estimates on something tending to $0$ is a lot less precise than basing them on something bounded by a constant.
big-O can fall prey to the same problems if the constant involved might change with some parameter in the equations.
but it is usually easier to account for these changes in the constant than it is to account for the change in convergence to $0$

Is there any good basic FOL textbook which covers the following things:
(1)FOL syntax
(2)A logical deductive system (Natural Deduction fitch style)
(3)Conventional and self contained
(4)Only requires pen and paper.

10:28 AM
@robjohn , makes sense. Minor detail; did you mean $\frac{f(u,h)}{{(hu)}^m}$?
This is the expression you use in equation 3 for little-o with two variables.

5 hours later…
3:06 PM
So I was teaching some divisors and somebody asked me this and I have never thought about this before (which is embarrassing but okay). So suppose we have a cartier divisor $D$ which is effective. Corresponding to this there is a line bundle $[D]$. In terms of locally free sheaves, this line bundle is $\mathscr{O}(D)$, and also the dual $\mathscr{O}(-D)$.
The sheaf $\mathscr{O}(-D)$ is what I thought were sections of the dual line bundle $[-D]$. But in the effective case, the definition of $\mathscr{O}(-D)$ implies we are looking for holomorphic sections. But for instance in something like a RS ($\Bbb{P}^1$), $D = p$, $[D] = \mathscr{O}(1)$ and $[-D] = \mathscr{O}(-1)$. But then the notion that $\mathscr{O}(-D)$ is the sheaf of sections of the line bundle $[-D]$ doesn't work. Am I missing something trivial?
Here by $[-D]$ I mean the dual line bundle to $[D]$, where $[D]$ comes from using the cartier divisor definition
No, god what nonsense am I blabbering

3:32 PM
Sorry everyone for the spam

1 hour later…
5:00 PM
I have $e^{-Px}\int Q'(x)\frac{e^{Px}}{P} \ dx$, with constant $P$ and $Q'(x) \to 0$ when $x \to \infty$. Clearly, expanding the integral with integration by parts will result in the integrated $e$ term being canceled by the $e^{-Px}$ in front of the integral so the whole thing will converge to $0$
but... how do I head in that direction more rigorously?

5:25 PM
you can use symbols other than $x$

5:52 PM
How do I know which symbol I should use for best results?

6:06 PM
use $x$

6:22 PM
i prefer to use two letter variables
like $xy$ and $yx$ etc

maybe klingon?

6:45 PM
im really getting annoyed and feeling unwelcome by all those questions of me that get closed for no good reason , even the ones with 7 upvotes

welcome to the club
you'll be getting your members card shortly
you can always ask in mathoverflow

thanks Yorch.
do you have similar expiences ?

well not always
yeah I have

yeah i posted it at Mathoverflow but they migrated here , and then merged it and then closed it.

:/

6:47 PM
the migration and merge i can understand , but to then close it with 7 upvotes and a bounty too :/

so a mod closed it

i guess
some mods seem to dislike me

as shaggy said, it wasn't me

its usually the same ppl , but other mods might have done the same dunno
its just strange if the users do not hate it , but the mods do

Oh I think what happened was kind of odd
you crossposted it?
and then it got migrated and merged with the crosspost
?

6:49 PM
all this " no context " is annoying , its math , there is no context. only homework , research or related topics have context anyways

and the version with the bounty dissapeared?

the mods are making the best of a fairly bad situation. it is an imperfect system.

we do agree re context but that is the way of the world.

the mods disagree with each other though

6:50 PM
yeah maybe they disagree , but it seems the closers always get their way

sure, but imagine being a closer
that's like an automatic L

after 7 years and 12500 rep things have not changed , to my huge frustration

I think the more rep you have the more you'll have to deal with closers tbh

it's gotten worse, i think because covid has more people trying to use the site as an answer service.
that kind of ruined it for everyone else.

there's like 350 questions a day
that's so low

6:52 PM
yeah leslie but my questions are CLEARLY NOT HOMEWORK

mick, i agree. it is the concept of one group of people ruining things for another group of people.

if only 350 questions a day , closing upvoted 7 times questions without mistakes , seems like a bad idea
350 is not much considering we have over 200 countries here

we agree. there are questions i haven't asked lately because i couldn't come up with anything resembling an 'attempt.' they are cooler about this on math overflow but i forgot my account name there.
MO also seems a little more focused on the status or situation of the questioner as opposed to the quality of the question. maybe that's unfair but it's the impression that i get.

I've never had a question on MO closed

its mainly a MSE problem yeah.
MO is cool but i see even less activity there ? less questions and answers ? ofcourse they have a higher level so that is a good excuse ... but is it growing or shrinking ?

6:56 PM
I think you can also get away with asking olympiad style stuff on MO but I'm not completely sure how much.
MO gets 33 questions per day

i dont know ... i never wanted to call those ppl by name , bc that would escalate and be ungrateful ( they answer alot often too ) but after all those years , perhaps that is the only way ...

MO has always been a very high quality site, although some disciplines are or were simply not represented there.

mods have changed over the time , but not necc improved.
bill dubuque was one the best. and some left because of the monica story

bill dubuque is/was eccentric but his answers were often of incredibly high quality.

Bill still uses the site

7:02 PM
i have essentially stopped answering questions. every now and then i will break down in need of some reassurance.

i feel the context argument should be removed entirely. math is math , no explaination needed. imo homework is ok too , but should be tagged and filtered

i don't think i have ever interacted with Bill, but somehow have the impression that he is a little abrasive.

i have formed the same impression although it is not all that different from some of the people i work with.

maybe i am just intimidated.
i have not been in a properly abrasive atmosphere for 1.5 years. i am losing my edge.

7:03 PM
i have seen him on sci.math and other math groups and he was always cool integer and good imo

cool integer?

we can meet at ben and nicks and i can make up for lost time.

his knowledge far outranks anything i could produce

didnt he got banned or left or changed his name ?

there was some kerfuffle. i don't remember the reason.

7:05 PM
his rep was at 1 once .. i guess that was a ban
disgracefull imo
putting his rep at 1 like a village idiot

life was never the same after my suspension, as behan once said
i hope that is a respectful picture of chuck norris.

there were times when you couldn't trust someone who hadn't been to prison.

i have been incarcerated once.
another country happy to see me leave
chuck can kill imaginary numbers.

i've stayed out of jail, only been pepper sprayed.

here is another example
3

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)

this is considered duplicate ... but the linked original is OLDER !??
the original OLDEST can never be the DUPLICATE ??
ok google told me there are only 195 countries.
so 350 question for 195 is like less than 1.8 per country per day.

7:23 PM
its gone silly.

@Leslie, further to yesterday’s discussion: proof for the sequence $(x_n)$ defined by $x_n=\cos n$ does not have a limit : since $\{\cos n: n\in \mathbb N\}$ is dense in [-1,1], there must exist subsequences of $x_n$ converging to 1 and -1 respectively that is $\limsup x_n=1\ne -1=\liminf x_n$ whence it follows that $\lim x_n$ does not exist.

on my site we sometimes close older questions as duplicates in favor of pointing people to newer and better specified ones, though that's because (barring some exceptions) questions on our site can't actually be answered, as in there is no concept of "OP has gotten the answer they wanted, no more is to be added"
here it may be similar though where the duplicate closure favored the one that wasn't a PSQ, but I don't know what Math.SE's policies are on actually retroactively dupe-closing questions especially if they're that old.

koro that is an interesting idea. it is possible to prove that lim cos(n) does not exist without relying on the density result. if the limit exists it has to satisfy a ton of trig identities.

7:46 PM
yeah we can without many trig identities. we can argue like this: suppose on the contrary that $\lim \cos n=l$, it follows from identity $\cos (n+2)-\cos n=-2\sin (n+1)\sin 1$ that $\lim \sin (n+1)=0$, and trig. identity $\sin (n+2)-\sin n=2\cos (n+1)\sin 1$ whence we get $\lim cos (n+1)=0$ which is a contradiction as it’s in violation of the identity $\sin^2 t+\cos ^2t=1$

Density is much harder.

Hi professor Ted. I think that density of $\{\cos n:n\in \mathbb N\}$ makes life easy :)
Leslie, I can’t stop thinking about your yesterday’s suggestion of using “continuous functions are completely determined by their values on dense sets” for proving density of some sets. I think that’s a very powerful application which I never thought of. Thank you so much for that :)

8:02 PM
I know/knew various proofs of density of $\{e^{in}\}$ in the unit circle, but none is simple.

@hyper-neutrino This is, more or less, correct. The goal of marking a post as duplicate is to provide a roadsign for future askers. The "best" version of the question should be at the root of the tree of duplicates. This can mean that an older question may be closed as a duplicate of a newer question.

@XanderHenderson Thanks for the explanation. Yeah, that makes sense, and I agree with it too; I've never considered duplicates really as "this question has been asked before" but rather "this question is the same or a subset of this other question, and we would like to point you to a place with more resources / that you should answer instead".
Granted, that usually is the same as "this has been asked before", but like in this case, not always.

8:21 PM
the problem is that duplicates rarely are

Heya folks

hi!

Been a while since I've hung around here. How go things?

8:56 PM
Hi
I have been working on the same problem for a week now
I feel I have found all but the last error
if you want to look at my post, here is the link:
2

Problem: Assume we have the following points: $(x_0,y_0), (x_1,y_1), (x_2,y_2), (x_3,y_3)$ where $x_0 = -3$, $x_1 = -2$, $x_2 = -1$ and $x_3 = 0$. Given the function $f(x) = Ax^2 + Bx + C$ find the constants $A$,$B$ and $C$ such that $f(0) = y_3$ and $$d = \sum_{i = 0}^{2} (f(x_i) - y_{i})^2$$ is...

9:18 PM
there is an exact procedure on one (the only) answer which you can use to solve the problem.

9:28 PM
You have chosen to disregard several of our direct suggestions. I’ve lost interest.

Heya Ted

Hi @Ted.

Hello chat

Heya Sayan

9:46 PM
Hi a @Balarka, Sayan, Rithaniel.

How's life?

Cooking today …

What're you cooking?

Hello, any general idea how to estimate the density of high dimensional function (like 60 dimensions) for example please?

functions with mass?

9:50 PM
Two new chicken recipes @Balarka … one extremely spicy, one extremely herby.
What is density of a function?

Haha, amazing. What herbs does the latter entail?

@copper.hat. You mean mass functions?

Fresh dill, tarragon, cilantro, mint (and garlic).

@Avra what does the density of a function mean?

Hmm, sounds familiar.

9:55 PM
I need to pick up some cilantro when I head back to the store next

@copper.hat. It's the distribution of data

I can give you some, Rithaniel ;)

You weren't joking when you said extremely herby

9:56 PM
@Avra that is a little vague. is there some measure involved?

@copper.hat. For example, given high dimensional data of size 60 (60 dimensions), how I can model that with density function and then find out low density areas of this fuction by projecting it to 2d manifold for example.

This is an entire field of study.

sorry, i don't know what you mean. maybe someone else has a clue.

I love coriander/cilantro in chicken.
Well, in many other things as well.

@TedShifrin, my questions is an entire field of study :0
@copper.hat. Thank you

9:59 PM
Topological data analysis.
Yup.

@TedShifrin. :0 So, how about gaussian functions? Can I use them to model high dimensional data?

No idea.

@TedShifrin. I mean, can I use a mixture of Gaussian functions to model high dimensional data please?

kinda a pretty wide open question

Probably, but you might need to practice with easier tasks to get up to where you can do that

10:04 PM
Hello, if we have to prove "A(n) iff B(n) for all n>1" , and we previously showed that B(n) is false for all n>1; does it suffice to prove that A(n) is false for all n>1?

@Rithaniel. Thank you. I did that with mixture of Gaussian functions, but I was looking for further hints if possible.

@pritchard That would cover the logical meaning, yes.

Thank you, just wanted to make sure

11:00 PM
Does anyone know where I can find an elementary proof (not using homology...) that there is no retraction from $D^n$ to $S^{n-1}$?

in dimension 2 i think there are other possibilities but i do not know a way for all dimensions.
if K-theory of spheres doesn't count as homology, you can use that. :)

11:20 PM
@sequoia13 Are you allowing Stokes's Theorem?
If so, see 3510 Day 44 in my YouTube lectures (linked in my profile). If not, I think there is no answer to your question.

Thank you!

11:36 PM
g--m-t-r to the rescue.

@TedShifrin. This is the first time I saw your YouTube Channel! You are famous Professor

LOL, just for YouTube videos? There are four texts, too. :)
@leslie Your code is beyond me.

i think it would be fair to classify ted as infamous.

@TedShifrin. I told you before you are blessed. Why not writing a philosophical book about math?

Despised despot.
Philosophical book?

11:44 PM
@TedShifrin. I am trying to remember a Professor who write amazig book about how he thinks about math for engineers! I forgot the book!
I read part of it 10 years ago

Every good text should show how the author thinks about the material in some novel way. Lots of bad texts.

@TedShifrin. It's more like philosophy behind concepts rather than a detailed math book
Probably Prof. Shannon

nice flow