« first day (1576 days earlier)   

12:04 AM
@PedroTamaroff @robjohn @anon @MikeMiller I want to check if the equation $3x^2+5y^2-7z^2=0$ has a solution in $\mathbb{Q}_p, p \neq 2,3,5,7$.

There are $\frac{p+1}{2}$ possible values of $3x^2$ and $\frac{p+1}{2}$ possible values of $7−5y^2$ for $x,y \mod p$. Then by the pigeonhole principle we have that for some $x,y$, we have $3x^2=7−5y^2 \mod p$, which then gives a solution.

Does this suffice, to show that there is a solution in $\mathbb{Q}_p, p \neq 2,3,5,7$, or do we have to apply also Hensel's Lemma?
 
@robjohn when flagging comments, who sees those? the Math.SE mods?
 
@MikeMiller Santa
 
mods and high rep users across the chat networks, not sure how it decides which among all the people see which flags
 
not chat comments
comments on main
 
oh
 
12:16 AM
I know that e.g. flagging questions or answers puts them into review queues for sufficiently high rep users (at least, it seems like it, since I frequently see questions there without close votes - where it just says it was e.g. flagged as off topic)
but there's no comment review queue, as far as I know
 
4
A: Probability of an integer being a prime

Daniel RFor the lower bound, note that the probability of selecting the first three primes is $$P\left(\omega \in \{2,3,5\}\right) = \frac14 + \frac18 + \frac1{32}=\frac{13}{32}$$ and the desired probability is obviously higher than that. To get an upper bound, consider the probability of choosing $2$...

How does he get $\dfrac{1}{504} ?$
 
by making a mistake, presumably
 
But he gets the answer I need, he must be very close
Do you see how to get the bound ?
 
if instead of $2^6$ it was $2^33^2$ it'd give you the same number
 
@PedroTamaroff you missed one very important thing related to my way: I broke some math rules to get my result ...
I wonder if you know how is that called.
 
12:22 AM
@Chris'ssis You should leave this place and never come back if such thing is true.
 
@Chris'ssis Which ones ?
 
@PedroTamaroff I think you should play with your kids and stop putting comments to my comments.
 
@Chris'ssis I don't have children. I am pretty young.
 
@Hippalectryon Look at the way I got the result here math.stackexchange.com/questions/1021647/…
How is called what I did there? You know?
 
@Chris'ssis So you basically confirmed you're that new disrespectful user?
I can't believe this.
You should leave this place and never come back if such thing is true.
 
12:26 AM
@PedroTamaroff I think you understand nothing from what I posted. My work is correct, and that broke has a certain meaning that is recognized in mathematics and used in some circumstances.
 
@Chris'ssis What is a "broke"?
@Chris'ssis Yesterday someone tried to mug me. He said "calculate the homology group of the sphere or I end you, boy." It was nice I knew how to do it.
 
@PedroTamaroff @Chris'ssis Oh please .... Don't start this again, both of you... I understand you both have something against the other, but please refrain from saddening that chatroom ... Let's get back to maths
 
@PedroTamaroff I remind you that some time ago you argued with me saying for an hour or so that $1/ 8$ is $0.25$. This is you.
 
Y'all need to get a room
 
@Hippalectryon I was talking about how math saved my life. How is this not inspiring?
@Chris'ssis Yes. I cannot count. But that's not new.
 
12:29 AM
@PedroTamaroff You know well enough what I'm talking about ...
 
@AlexanderGruber I need your help with something.
 
@PedroTamaroff Yessir
 
@AlexanderGruber Facebook-.
 
@Chris'ssis I'll take a look at it tomorrow
It's 1:30 AM here
 
@Hippalectryon Do you see what I did there? The special thing? OK
 
12:32 AM
@Chris'ssis I don't see it right away, but once again, i'm tired
 
@Hippalectryon I'm too and possibly I'll be called tomorrow for another interview. And it's also late here but ...
 
Hope it goes well :-)
 
@Hippalectryon In all places I worked people respected me profoundly.
 
@Chris'ssis oh cool where are you working?
 
@Chris'ssis I've ever worked :-) (as in, real life work)
 
12:34 AM
@AlexanderGruber at the moment I attend some accounting and tutoring (much more accounting than tutoring).
 
Ah neat, is that corporate tutoring or uni students?
 
@AlexanderGruber In general some high school kids, easy stuff.
 
@Chris'ssis I wish I'd known that last summer. I was teaching this applied calc course with all manner of integration, and >50% of the students were accounting
you probably could have helped me write good relevant examples. I don't know anything about accounting.
 
@AlexanderGruber I'm not sure I got your point. Integration in accounting? I never did such a thing.
 
@Chris'ssis Just stuff to relate the material to something relevant to them, even if indirectly
Even if it's understanding another part of their business or some relevant finance thing, I just don't know any accounting terminology whatsoever
 
12:37 AM
@AlexanderGruber If you're interested, I can give you some of my e-book I studied, but you need to translate them.
 
@Chris'ssis from which language?
 
@AlexanderGruber Romanian. I have somewhere these books I studied, both hardcopies and e-books.
 
Yikes :p I do not know Romanian.
 
@Hippalectryon You're right that his bound is better than the one your question requires but it's just so curious to me that the 209/504 thing looks so nice. It should be easy to get, considering how easy the lower bound was.
 
@MikeMiller mhm
 
12:39 AM
@Chris'ssis thanks though, I don't know if I'll ever be teaching that same course.
it does seem interesting but I have a lot of independent projects. lately i've been trying to simplify.
 
@AlexanderGruber If you need help, just ask and if I can help, I'll do it.
By the way, I need to see where I put my e-books from my all years.
 
@Chris'ssis I'm interested
 
@Hippalectryon In some new integrals? :D
 
@Chris'ssis That also. I meant, in your ebooks
 
@Hippalectryon OK. I need to find them, I also need to send them to other kids that study the financial accounting (although they are not that interested in learning ...)
 
12:45 AM
@Chris'ssis You don't have my email, do you ?
 
Anyway, there is a big difference between learning and study.
 
Ah nvm I can't send pm on SE, so I can't give my email privately :/
 
@Hippalectryon No, I don't have it, but I cannot afford to ask for anyone's e-mail.
 
@Chris'ssis I would just thinking it would have been easier to send ebooks as files
 
@Hippalectryon This is not a problem, I only need to find the pack.
 
12:47 AM
Or, are they all available online ?
Oh Ok
 
@Hippalectryon No, they aren't available online (as far as I know).
@Hippalectryon tell me 3 numbers of pages ... and then wait a bit
 
@Chris'ssis 3 numbers of pages ? uh... 175, 32 and 537 ?
 
537?
@Hippalectryon tell a smaller one
 
Anyone interested in a field theory problem! I posted an answer which I suspect is faulty. @anon @PedroTamaroff
 
1:02 AM
@SwapnilTripathi "union of field extensions" is ill-defined. even if we put the two field extensions inside a common larger field, the set-theoretic union is not going to be a field. one speaks of the join of two field extensions within a lattice of field extensions ordered by inclusion.
 
@Chris'ssis Sorry but I really need to go to sleep if I don't wanna turn into a zombie. If you find them, please ping me. When I'm back SE will notify me of your ping.
 
@Hippalectryon 3 min
 
@Chris'ssis Oh ok then
 
@Hippalectryon Integrals will eat up your brain.
 
1:04 AM
Why don't you learn about proper analysis?
Metric spaces, real analysis, measures?
 
@PedroTamaroff I'm in 'college'
 
Then you can look at a much bigger and interesting class of integrals.
 
Like singular integrals, Dirichlet integrals, complex integrals.
 
@Chris'ssis :O
 
1:05 AM
@Hippalectryon I'm not sure the number of the pages can be seen.
 
They can't be seen :/
 
@Chris'ssis Why not a picture of yourself?
That'd be interesting as fudge.
 
@AlexanderGruber I not only studied those books but I also corrected some, and also corrected some exams tests. It was often met during my exams to get not a good mark because the tests were wrong.
 
@anon: Could you please help the OP? He has posted a couple of questions to get to some conclusion. But he hasn't been able to do so.
 
@Chris'ssis how different is it from actuarial science?
 
1:06 AM
@SwapnilTripathi re: your answer, to define A(B) both A and B need to be in some larger field, but proving A and B embed into a larger field is the entire point of the question.
 
Oh, I get it now! @anon
 
@Hippalectryon I wanted to open those books precisely at those pages ... I studied things I didn't play with them.
 
@PedroTamaroff A picture with me? You need one to do some voodoo on it? :-)
 
1:10 AM
@Chris'ssis I don't hate you Chris. I only hate integrals.
And I cannot vodoo integrals.
 
@anon: Thank you for the help. I'm off. See you! :)
 
@PedroTamaroff I don't hate people either.
 
@Chris'ssis Well, have a good night
 
@Hippalectryon Good night.
 
away dreaming of integrals
 
1:11 AM
@Chris'ssis What about Stalin?
 
lol @Hippa
 
@PedroTamaroff People are not dangerous, but some ideas that could bring to life horrible monsters ... The greatest battle is the battle in the ideas field.
 
You can do voodoo to people you like.
Get a picture of someone you like, draw a smiley face on it, and give them a balloon.
 
@AlexanderGruber Well, if I refer to en.wikipedia.org/wiki/Failure_mode_and_effects_analysis, then it's a big difference. I didn't study it at school al all.
@AlexanderGruber it's at least as hard as the integrals and series I do if you have a deep understanding of it. It's the real analysis of manufacturing processes. Well, just a way of calling things, the way I like. Besides the fact that you need to know all these processes (almost perfectly) you need to be able to see things in the future and prevent any kind of failure.
@AlexanderGruber you need to think of things that no one, exactly, no one did it ever and come up with preventive and detective solutions for all of these potential risks within each of the proceeses. It's amazing to work in this area since the complexity is absolutely crazy. You're both a mathematician, but at the same time a very good detective.
 
No detective's gonna stop me.
Also, Hi everybody!
 
1:24 AM
@AlexanderGruber Well, you have some charts to assess the classical variables used for the assessment of the severity, occurence and detection, but at the same time you need to know how to combine them with many other variables you take them from the spot such the you have the real situation of things every day.
 
@KarlKronenfeld HARRO
 
@AlexanderGruber what you do may produce huge savings to your company, that is many many millions of euros. In general you need to work like 2 years in the engineering department before getting the job. Then you need like 2 other years to be a good (not that good though) specialist in the area.
Again, its complexity is phenomenal ... that's why I love it that much ...
@AlexanderGruber newbie will never understand your work in the first 2 years, but he needs to accept that your job is very important.
Well, there are some cases when lots of engineers do not realize the importance of the technique even if they are in the company for many years. There is a stuggle for that, and many trainings. You need to have a deep understanding of things, you know almost all the cells of the plant.
 
Hello!!

I want to check if the field extension $\mathbb{Q} \leq \mathbb{Q}(a+ia)$ is normal. ($a \in \mathbb{R}$ with $a^4=5$)

I have done the following:

$Irr(a+ia, \mathbb{Q})=x^4+20$

$x^4=-10 \Rightarrow x^2=\pm 2ia^2 \Rightarrow \pm \sqrt{2i}a, \pm \sqrt{-2i}a \notin \mathbb{Q}(a+ia)$

Therefore, $\mathbb{Q}(a+ia)$ is not a normal extension of $\mathbb{Q}.

Is this correct?? Have I found the solution of $x^4+10=0$ correctly??
 
@AlexanderGruber On the other hand you may have 5 years experience and your outcome to be very poor. In my opinion, one needs to be gifted to get such a job and crazy lover of the details. Then you need to connect together all the information you get every day, that is huge!
 
@Hippalectryon What's the probability that one urn holds at least 2 balls? Extend that case to the fact you have $n$ options of such
 
1:33 AM
@AlexanderGruber I might talk a lot here on the topic, but the requirements are extraordinary ... (I cannot describe that by words)
 
@Studentmath: You're supposed to be asleep!
 
@Hippa alternatively, the probability that no urn holds two or more balls - precisely the same question but the different phrasing might give rise to an idea
@Ted There's a huge storm outside, woke me up..
We aren't used to these things
 
@TedShifrin Maybe you'll have something to say to this guy.
 
But yeah, I am going back to sleep :P it's 4 AM soon, waking up at 6.. g'night once again
 
@AlexanderGruber The thing that breaks hearts at most is the moment when you go to the interview and see people there that they believe they know so much about this technique, but they know almost nothing, you depend on them, they assess you.
I reached a very high level in this techique, I also developed it a lot. I'm aware of that very, it is very.
Anyway.
 
1:38 AM
@TedShifrin Hi!! Could you maybe give me some hints how to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$??
 
I need some sleep.
 
@MaryStar what's your definition of $\Bbb F_{2^2}$?
 
tries to come up with an exotic definition
 
the field of least cardinality whose underlying additive group does not have a faithful smooth action on the circle?
 
@anon $\mathbb{F}_{p^n}$ is the only field with $p^n$ elements.

$a \in \mathbb{F}_{p^n}$ then $a^{p^n}=a$
 
1:54 AM
I can't think of any actually interesting ones, @KarlKronenfeld
and tragically, the one I gave above is not true
 
it does actually have a faithful action, right?
 
yeah
quite obviously so
 
making sure I wasn't blundering
 
even one that acts by isometries
now if you demand orientation preserving isometries, we've got a deal
 
1:57 AM
@MaryStar if a has degree 2 over F2 then what is the size of F2(a)?
 
@anon $2$??
 
well, no
 
the other unique field with 2 elements
 
F2 contributes a couple of elements to F2(a)
then you have a and whatever other crap tags along
 
@MaryStar so F2(a) is bigger than F2, but has the same number of elements as F2?
@MaryStar how about this. Say V is a vector space of dimension d over a finite field with q elements. In terms of d and q, how many elements does V have?
 
2:06 AM
@DanielFischer Don't worry, I think the problem is simpler than I thought
 
@anon I got stuck right now... How can we find it??
 
@KarlKronenfeld Any news?
 
@MaryStar V has a basis, so we can write everything with coordinate vectors, having d entries. how many ways are there to write a coordinate vector with d components, if each component has a scalar from a field of q elements?
 
@PedroTamaroff Thanksgiving break has started here, so some sleep occurred
 
you've let on that you're a student
finally, I have info to feed my armada of spies and assassins
 
2:19 AM
not necessarily, just very likely
 
true; but it means I can have them focus their attention away from the department store managers I've had them hunting for a while now
 
how'd you know?
 
@MikeMiller raider nation armada reporting sir.
 
Assassin's division };-)
 
2:32 AM
I am having trouble proving the following: Let n be an integer greater than 1. $(1+x)^{1/n} < 1 + \frac{x}{n}$ if $-1 < x < 0$ or $x > 0$. I am not entirely sure where to start. I've been stuck on this problem for a while now.
 
Thanks for sharing that wiki article @KarlKronenfeld :D
 
@anon So, the basis is of the form $v_1, v_2, \dots , v_d$ and we want to write the elements of a finite field with $q$ elements using the basis. Is this correct??
 
you're looking at how to write vectors using the basis and scalars from the field, not looking how to "write the elements of a finite field"
 
3:12 AM
@anon Could you maybe give me a specific example??
 
@MaryStar pick any values for the numbers and make your own example
Oct 21 '13 at 5:03, by anon
do not wait for math to penetrate you. you must go out of your way to penetrate it.
 
I am not so sure I like that phrase
oct 21 '13 chat looks like a nice place.
 
3:31 AM
yeah, well, the word choice was there before my comment, so
 
good point.
 
@anon Smooth.
 
3:46 AM
Ah, the good'o days...
 
4:24 AM
Hello Professor @TedShifrin
 
4:41 AM
btw @PedroTamaroff thanks for getting the Professor to come to the chatroom :-)
 
I cannot remember when that happened.
Let's see the history.
OK, it was on August 21st, 2013.
 
LOL @Pedro
 
@skullpatrol Oh, it was before.
 
May 29, 2013 @4:05 AM :-)
 
4:46 AM
Hehe, I remember when Mariano was making caramel candy. I had some.
 
Oh, it must be 4 AM UTC.
I was confused why everyone was awake at such an hour.
 
4:59 AM
The first message:
Aug 3 '10 at 15:36, by Chao Xu
I want some system that can help with collaborative mathematical problem solving.
 
5:13 AM
@Pedro Please get Mariano to make candy for me.
 
@MikeMiller I haven't talked to Mariano in a while.
I hope to see him more often next semester.
 
My request stands.
 
5:45 AM
Me too :(
 
 
2 hours later…
7:40 AM
Find the matrix of the rotation in $R^3 $ through the angle $\alpha$ around the vector
$(1,2,3)^ T$ .
We assume that rotation is counterclockwise if we sit at the tip of the
vector and looking at the origin.
 
7:52 AM
I think the answer is $
\begin{pmatrix}
1 & 0 & 0 \\
2 & 0 & 0\\
3 & 0 & 0 \\
\end{pmatrix}
$$
\begin{pmatrix}
\cos \alpha & -\sin \alpha& 0\\
\sin \alpha & \cos \alpha& 0\\
0 & 0 & 1 \\
\end{pmatrix}
$$
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0\\
0 & 0 & 0 \\
\end{pmatrix}
$
Am I right?
 
8:31 AM
il y a 2 ans et 1 jour que j'habite sans lui
il y a 2 ans et 1 jour que je ne le vois pas
et bien que je n'ai été pas hereux, j'ai appris à vivre sans son amour
 
9:11 AM
@KajHansen "Math is torture..." :D
 
10:02 AM
@skullpatrol hi pal
 
if you sample with replacement from n items and you want to estimate the mode
is taking the mode of the sample a good way?
 
10:19 AM
@Twink hi pal :-)
How could it not be @user2179021?
 
10:33 AM
You like that? @skullpatrol
 
@skullpatrol what was the question?
anyone got any ideas about math.stackexchange.com/questions/1039829/… ?
do you think it is hard?
 
It is an interesting POV @KajHansen
 
10:49 AM
I could upgrade it to MO but I suspect @robjohn could solve it in 5 minutes :)
or @user21820
 
Greetings
I just had a perfect answer downvoted ...
1
A: How Prove this integral is diverge $\int_{0}^{1}\dfrac{dx}{\ln{x}\ln{(1-x)}}$

Chris's sisNote the simple fact the integrand is positive and $$\int_{0}^{1}\dfrac{1}{x\ln{x}}\cdot \underbrace{\frac{x}{\log(1-x)}}_{\large\text{near 0 it behaves like $-1$}}dx\longrightarrow \infty$$ Q.E.D.

@robjohn MSE leading team do not care at all about the downvotes I receive every day? I mean what is the value of the users on this site because it seems to me no one cares about many of them. Targeting downvoters make the rules here.
 
hi @Chris'ssis
I see integration is your one true love
I must learn how to rephrase all my questions in terms of tricky integrals :)
actually I think I can see how to do it. Take logs and replace the sum by an integral :)
@Chris'ssis shall I do it? :)
 
@robjohn The reson for that I left my last job was because of responsability, no one cared about the responsability. I hate any place where no one assume a responsability (and in this case is about the shitty targeting downvoting).
I mean anytime one creates an account and then use it to hugely downvotes the people he wants. Why are these accounts allowed to exist? This is the main question.
 
11:07 AM
@Chris'ssis Quick can you help me switch order of integration?
 
@N3buchadnezzar No
 
@Chris'ssis it's an interesting question whether people who just downvote get punished by the SE system
I suppose they could have a role
 
No, they do the rule here. I mean you create an account and then you do what you wanna do. I mean you can spread shit everywhere.
 
Haters gotta hate
 
11:08 AM
@Chris'ssis First of all, no one monitors people's accounts. If there is someone downvoting, then you need to let us know. I know you have mentioned this in the past, but not recently. Next of all, I have just looked and as far as I can see, there does not seem to be evidence of targeted downvoting. However, I will pass it up to those who can look deeper.
 
@robjohn I take at least 2 downvotes every day. This cannot be a coincidence.
 
@Chris'ssis Yes, you have a lot of down votes, but they are not from a single user or even a few.
 
$$ \int_0^2 \int_x^{\pi x} f(x,y)\,\mathrm{d}y \,\mathrm{d}x$$
 
hi @robjohn
 
@robjohn I know, they must come from some specific accounts. Besides that there are some users that might downvote me for other reasons.
 
11:10 AM
@Chris'ssis I will look to see if any of these people seem to be from sock puppet accounts, but that is as far as I can look. For more information, I need to kick things upstairs.
 
@robjohn OK, thank you.
 
@Chris'ssis people's reasons for downvoting, we cannot control, but we can note patterns and take action on those.
 
@robjohn Yeah, perfect, I totally agree with that. I only refer to those sick patterns.
@user2179021 I also love the series and limits.
 
Have you ever heard the saying "take it with a grain of salt" @Chris'ssis?
 
@skullpatrol No
@skullpatrol What's the meaning?
 
11:14 AM
It means "allow for exaggeration"
Try googling it for a more elaborate explanation :-)
 
@skullpatrol Ah, I see now. :-)
 
Perhaps somebody has an exaggerated sense of self-righteousness.
 
@skullpatrol you refer to the downvoter?
 
yes
 
@skullpatrol or there someone needs some serious medical care. Especially when you're obsessed with someone.
 
11:20 AM
Just take their actions "with a grain of salt" recall what you come here for is to do math, not politics :-)
 
Watson is auctioning his nobel lol
 
@skullpatrol Do you think I'm wrong when I say that these accounts should be deleted? I mean you have an account to downvote people, not to do math. Why should such an account exist?
 
don't waste your energy on their mental problems, right?
 
@robjohn I remember that you also were downvoted some time ago, almost daily if I'm not wrong.
 
Crick too @UserX?
 
11:29 AM
@skullpatrol well, after you post some answers and questions on this site, you look at things a bit differently. I mean you work on these questions and answers, you consider them precious, it's your effort there. I mean if you posted more answers and questions then you'd have known what I meant.
 
Don't post your most precious results, @Chris'ssis keep them for your book :-)
 
@skullpatrol Ah, that's sure. :-)
 
My point is don't get too emotionally involved with these trolls, they are lonely people just looking for attention. @Chris'ssis
 
@skullpatrol nah
 
If you don't pay any attention to them @Chris'ssis they will get bored and go away :-)
 
11:38 AM
@skullpatrol Yeah, that's good to know.
 
@skullpatrol crick's medal was already actioned
But he is dead since 2004. Watson is the only living nobel winner to auction it.
 
Wow @UserX I once read they didn't give any credit to the grad student who actually made a key observation in their lab that lead to the discovery
 
@skullpatrol long story
 
@skullpatrol I don't know if I told you ever, but you're the most mysterious person here. Who are you? :-)
(just asking)
 
skull pa troll hunter @Chris'ssis :-)
 
11:44 AM
@skullpatrol lollllllllll :-)))))
 
Given two reals $a,c\in\mathbb R$, is there a formulaic way to obtain a rational $q\in\mathbb Q$ with $a<q<b$?
 
@robjohn I attached an awesome bounty to an awesome question. :-)
 
@MikeMiller I assume you are asking about flags on question comments. If you are asking about flags on chat comments, ask again. Custom flags on question comments are seen only by math mods.
@Chris'ssis I have been working on an answer that incorporates a couple of old answers.
 
@robjohn Great! No hurry, I let that until the last day, last hour, and even more.
 
11:49 AM
@Chris'ssis the problem is that I have had little time for answering questions the last few days. I hope to get back into things this weekend.
 
@robjohn I only wanted to be sure you are aware of that bounty.
 
Hello, I want to prove the continuity of $x^2$
how to do to find $\delta$ when $|x^2-x_0^2|\leq \varepsilon$
please
@robjohn please
 
12:06 PM
@Vrouvrou You mean how do you find a $\delta\gt0$ so that when $|x-x_0|\le\delta$ you have $|x^2-x_0^2|\le\epsilon$?
 
yes
 
Note that $x^2-x_0^2=(x-x_0)(x+x_0)$
 
yes i do this after that i stop
@robjohn please
 
so you need $\delta(x+x_0)\le\epsilon$, right?
 
yes
 
12:13 PM
@Vrouvrou You also know that $x$ is close to $x_0$; that is, $|x-x_0|\le\delta$
So say that we restrict $\delta\le|x_0|/2$, which is okay, since we are only looking for small $\delta$
 
we can do this ?
@robjohn ok
and then ?
 
@Vrouvrou certainly. We just say "let $\delta\le\min\left(|x_0|/2,\frac23\epsilon/|x_0|\right)$"
Then $|x+x_0|\le\frac32|x_0|$ (triangle inequality)
Therefore, $|x^2-x_0^2|=|x-x_0||x+x_0|\le \frac23\epsilon/|x_0| \cdot \frac32|x_0|$
 

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