Mathematics - 2012-06-18 (page 1 of 3)

Peter Tamaroff

00:00

@Eugene Hahahha remember I'm hitting 10k sooner or later.

Eugene

come at me bro

Peter Tamaroff

@Eugene Oh, I'll come. Wait, I mean...

Dear an-interesting-multivariate-calculus-problem

Eugene

i should just rename every problem as "an interesting problem"

Peter Tamaroff

@Eugene Malasya you said? Malasya it'll be.

Eugene

:4995001?

Peter Tamaroff

00:06

@Eugene That's where you live, right?

Eugene

@PeterTamaroff i should start with your questions actually

"an interesting number theory problem"

Peter Tamaroff

@Eugene Hahahaha

Eugene

"another interesting problem"

Peter Tamaroff

@Eugene "Yet, another interesting problem."

Eugene

@PeterTamaroff "a mind-blowing problem"

Peter Tamaroff

00:10

@Eugene "Seriously, I'm puking rainbows right now. Readers, beware."

@Eugene Oh!

One thing.

$$\eqalign{
& \left( {{{11}^8} - 1} \right)\left( {{{11}^8} + 1} \right) \equiv 0\bmod 17 \cr
& \left( {{{11}^8} - 1} \right)\left( {{{11}^8} + 1} \right) \equiv 0\bmod 17 \cr
& {11^8} \equiv {121^4} \equiv {2^4} \equiv - 1\bmod 17 \cr
& {11^8} + 1 \equiv 0\bmod 17 \cr
& {11^8} \equiv - 1\bmod 17 \cr
& {11^{104}} \equiv {\left( { - 1} \right)^{13}}\bmod 17 \cr
& {11^{104}} \equiv - 1\bmod 17 \cr} $$ This is OK right?

Eugene

what's the question???

well since i don't know the question anyway i'll just say yes!

Peter Tamaroff

@Eugene Sorry!

I had to prove that $11^{104} \equiv -1 \mod 17$

Eugene

oh

trivial

Peter Tamaroff

using The Euler/Fermat Theorem.

Actually, since Burton doesn't talk about reduced residue classes, it was only $a^{p-1} \equiv 1 \mod p$ for $p \not\mid a$

Eugene

$11^{16} \equiv 1 \pmod{17}$ no?

Peter Tamaroff

00:15

@Eugene Jah.

Eugene

so this is simple then no?

$6 \cdot 16 = 96$.

hi @Gigili

Dylan Moreland

You're doing work on the main site, @Eugene.

Eugene

@DylanMoreland huh?

Peter Tamaroff

@Eugene I used that $7 \cdot 17 = 119$

@Eugene You're answering stuff.

Gigili

Haha, the answerer refuses to answer the question even when I'm asking for the final answer while looking at him directly in the eye.

Eugene

00:19

@PeterTamaroff $11^{104} = 11^{96} \cdot 11^8$

Gigili

Hi, Eugene.

Eugene

@Gigili which question?

@PeterTamaroff where?

@DylanMoreland sorry i don't understand what you mean.

Jordan

I almost have as many profile views as reputation

Gigili

This thingy

Peter Tamaroff

@Eugene And $11^8 \equiv -1 $

Eugene

00:22

@Gigili yeah. it happens though. yesterday the OP accepted an answer that didn't answer his question. this was pointed out by qiaochu.

@PeterTamaroff yup. so it's simple

Dylan Moreland

@Eugene I just see a lot of number theoretic stuff on the main site, usually bumped by you. It's nice, that's all.

Peter Tamaroff

@Eugene Ha, well, I'll develop that slickness eventually.

Eugene

@DylanMoreland oh yeah, thanks! i decided to dispose of the modular form questions. for some reason they aren't as well received as my elliptic curve answers. i cited my paper in one of my answers though so that was cool.

Dylan Moreland

Fewer people like modular forms, I think. A little more inscrutable.

Eugene

@PeterTamaroff it's just multiplication. nothing slick about it. for slickness read rudin. it's almost like he dipped analysis in oil.

Gigili

00:25

@Eugene Well, yes (+1). This situation will not stand. I've got an idea, we should make a list of these OPs and suddenly they all disappear mysteriously. evil grin

Dylan Moreland

As an object of interest. Obviously they're both ridiculously complicated.

Peter Tamaroff

@Eugene LOL

Eugene

@Gigili this is a little scary. as long as i don't disappear.

@DylanMoreland indeed! i spent more years learning modular forms though. i didn't have the algebraic geometry to understand elliptic curves until recently

i somehow think modular forms are more accessible as all you really need is complex analysis up to cauchy's residue theorem to understand them (on a surface level at least)

Dylan Moreland

Also true.

Eugene

but proving things about modular forms is pretty ugly at times. very technical.

Dylan Moreland

00:30

I guess that's what I mean.

Eugene

yeah... i guess you're right. elliptic curves really have much slicker proofs.

Dylan Moreland

I used to not appreciate congruences at all, either.

Eugene

and you do now?

Dylan Moreland

I think so.

Eugene

they do have very nice local-global properties. i like hensel's lemma in that respect.

Dylan Moreland

00:33

I was reading Swinnerton-Dyer's paper (there's some recent thread about this) in Antwerp III and said that I didn't see why any of this was happening and Matt gave a very convincing speech.

Eugene

was it this thread?

Dylan Moreland

I guess the main point was that this is how everyone he knows proved the major theorems of the past 20 or so years. That seems like enough.

The first person who started noticing these things was probably nuts. But it makes sense to do it now.

I wish robjohn were here; I need to find a place to eat in LA.

Eugene

@DylanMoreland you're in LA now? cool. i thought you were at the fields institute still.

@DylanMoreland doesn't this describe most of number theory? =)

Dylan Moreland

@Eugene Indeed.

@Eugene That was just for two weeks. Now a week here for Hida's birthday conference.

Eugene

i remember thinking this when i read minkowski's lemma to prove the finiteness of the class number. i didn't see how he would have thought of it.

@DylanMoreland ah

something that always comes up amongst the mathematicians i talk with is the "back in time" discussion

the one where we ponder "if i were to go back in time would i be able to prove the theorems in field X"

Jordan

00:52

god I suck at math

I spent like four hours on that problem yesterday and it takes like 10 seconds to solve

Peter Tamaroff

01:08

@Jordan God invented the integers. We did everything else. Don't bother him.

Peter Tamaroff

01:30

@Eugene Isn't this obivous:

anon

1+1=2

Peter Tamaroff

If $S(n)$ is the number of squarefree divisors of $n$ then $$S(n)=\sum_{d \mid n} |\mu(d)|=2^{\omega(n)}$$ where $\omega(n)$ is the number of distinct prime divisors of $n$.?

@anon HA, HA. =)

anon

Seems clear to me.

Peter Tamaroff

@anon Yeah, same here.

@anon Maybe it can try to prove that

If $n=p_1^{k_1}\cdots p_n^{k_n}$

And $f(n)$ is multiplicative

Then

anon

$f(n)=f(p_1^{k_1})\cdots f(p_n^{k_n})$?

Peter Tamaroff

01:35

Nah! Wait!

anon

:jeapordytheme:

Peter Tamaroff

$$\sum_{d \mid n} f(d) \mu(d) = \prod_{k=1}^r (1-f(p_k))$$

Eugene

1+1 = 2 is not obvious!

Peter Tamaroff

@Eugene 3 volumes!!!

anon

@Eugene I was thinking of following it up with a Russel joke, but decided not to.

Eugene

01:36

@anon i would like to hear it

anon

I didn't actually have a Russel joke, I just wanted to have one.

I have that problem a lot.

Jordan

The thing I like about programming is that there are many ways to get a correct answer and they can all functions very diffeently, it really just depends on your creativity. In math there is a single way to get the answer, maybe two but it all depends on memorized knowledge of previously learned methods. No creativity.

Eugene

the proof that 1+1=2 apparently

anon

@PeterTamaroff I think the argument for that is similar or parallel to the # of sqrfree divisors one, but I don't think you can use the one to establish the other.

Peter Tamaroff

@anon Oh, I don't think it does.

anon

01:38

Instead, what I would do is prove that for $d|n$, $\mu(d)\ne0$ if and only if $d|n'$, where $n'$ is the radical of $n$. (You have FTA at this point right?)

Eugene

@anon at least he didn't go "wait! listen!"

Peter Tamaroff

@anon Radical?

anon

Eh, nevermind.

Peter Tamaroff

@anon There's no turning back.

anon

Okay, the radical of $\prod p_i^{r_i}$ is $\prod p_i^1$

Peter Tamaroff

01:41

@anon That wasn't hard, wasn't it?

anon

Anyway, I would just decompose the set of divisors $d|n$ as $d_1\cdots d_k$, where $d_\ell\in\{1,p_\ell,\cdots, p_\ell^{r_\ell}\}$

Peter Tamaroff

@Eugene I gotta say I really like the excercises in Burton's book.

anon

I guess the moral here is: prove it combinatorially.

Peter Tamaroff

@anon I'll give you a proof in some days, since I haven't gotten to that chapter.

But I just found the first "obivous" one and it called my attention.

anon

the second one is obvious too if you work with Euler products enough

Eugene

01:44

@PeterTamaroff i've been telling you that it's the better book to learn elemNT from.

Peter Tamaroff

@Eugene *best

Wait

Consensus

AnNT
AgNT
EmNT

Jordan

I just realized when typing someones name you can hit tab to select the first option

anon

You can also press tab more than once and it will cycle through the names.

Eugene

someone please upvote this.

Peter Tamaroff

@Eugene I think this is the kind of work Dylan says you're doing.

Eugene

01:47

thank you. for the upvote.

i've just been trying to clean out unanswereds from the list of modular forms and elliptic curve questions. i'm running out though.

Peter Tamaroff

@Eugene Well, good for you then.

I'd do the same with "my" tags, but I'd be crazy.

Eugene

@PeterTamaroff i tried cleaning out the algebraic number theory ones as well but that's tough.

@PeterTamaroff is this a general form solution?

Peter Tamaroff

@Eugene I think you should know this

A line can be written as the "general form", the "slope, y intercept" and the "point slope".

Ie

$ax+by+c=$
$y=mx+c$
$y-y_0=m(x-x_0)$

Those are just pedagogical words.

Eugene

i've never heard of the term before... since i learnt math in a DIFFERENT LANGUAGE

Peter Tamaroff

@Eugene Maybe I should've written

Not should as "you should've known" but

But should as "it will make things clearer if you know that".

lol

Eugene

01:55

bah!

lol

Peter Tamaroff

@Eugene What is that different langauge? (I did too, hombre)

Eugene

malay

Peter Tamaroff

@Eugene It is related to what "great" language?

Eugene

well holland got eliminated. why do they always disappoint me?!

@PeterTamaroff no great language. barely anyone else in the world speaks it.

Peter Tamaroff

@Eugene Cool. Keep it alive.

Eugene

01:59

@PeterTamaroff i don't plan to.

Peter Tamaroff

@Eugene Why not?

Eugene

@PeterTamaroff i'm not really fond of my home country

Peter Tamaroff

@Eugene Oh. I really don't know much about Malasya, though. Why?

Eugene

@PeterTamaroff it's a racist country full of corruption and oppression. does one need more reason than that?

Peter Tamaroff

@Eugene One of the three suffices.

Eugene

02:02

@PeterTamaroff cool.

Peter Tamaroff

My country is very corrupt and deluded, that's why I'm not fond of it.

I think very few people are fond of their home country.

@Eugene What is the current "democratic" state of Malaysia?

leo

hi there!

Eugene

@PeterTamaroff it's malaysia and it's not democratic at all really.

Peter Tamaroff

@Eugene Why so?

Eugene

@PeterTamaroff just rigged elections.

Peter Tamaroff

02:14

@Eugene It is unfair I'm watching a political programme which is mocking our President.

@Eugene But you aren't living there right?

Eugene

@PeterTamaroff nope.

Peter Tamaroff

Malaysia has had one of the best economic records in Asia, with GDP growing an average 6.5% for almost 50 years. The economy has traditionally been fuelled by its natural resources, but is expanding in the sectors of science, tourism, commerce and medical tourism.

The country is multi-ethnic and multi-cultural, which plays a large role in politics. The government system is closely modelled on the Westminster parliamentary system and the legal system is based on English Common Law. The constitution declares Islam the state religion while protecting freedom of religion. The head of state is …

Wiki

Eugene

i wish this were true.

Peter Tamaroff

Oh, OK.

@Eugene Is it a theochracy?

leo

@Eugene I thought that you where from Canada

Peter Tamaroff

02:18

Is the Islam influential there?

Eugene

@PeterTamaroff sort of a theocracy

@PeterTamaroff this is an example of information control in malaysia

Peter Tamaroff

What does

Kalau rasa tak bersih, bubar aje la DUN Selangor, Penang, Kedah dan Kerlantan.Ada berani...
mean?

robjohn

@Eugene 4 points yesterday ;-)

Eugene

@robjohn the difference between us?

robjohn

@Eugene yes, from the tag wiki edits (you can't get those points when over 20K)

Eugene

02:24

@robjohn yeah! modular forms tag-wiki for the win!

robjohn

@Eugene :-D

Eugene

@PeterTamaroff it's not important. it's just gibberish

@robjohn 14 points so far today though.

Peter Tamaroff

@Eugene I'll google translate and get something that might not be what is being said.

@Eugene You're very true to your field =). Respect!

robjohn

@Eugene 14? I saw 10 today, or are you including the 4 from yesterday?

Eugene

@PeterTamaroff it's really just nonsense. translating it would be quite meaningless from me as well.

@robjohn well in the weekly's we are 14 pts apart

robjohn

02:26

@Eugene yes

Eugene

@robjohn bah. now it's 4

@PeterTamaroff i just don't know anything outside of it...

Peter Tamaroff

Oh, God, I'm done with epistemology.

Eugene

i don't like philosophy.

Peter Tamaroff

@Eugene Me neither.

I mean, I like to discuss and stuff, but words get boring.

Eugene

yeah. then don't take it

Peter Tamaroff

02:32

@Eugene It is compulsory. And it ends tomorrow if I get 55% or more.

Eugene

@PeterTamaroff hahaha

Peter Tamaroff

I already have a 85% so I just need to average 70%.

Eugene

you should just do well do grad schools don't think poorly of you

Peter Tamaroff

@Eugene Well, I have 95% in physics and 100% in anal, so I'm cool =P

Eugene

@PeterTamaroff it figures you'd be best at anal

Peter Tamaroff

02:35

@Eugene Well, it seems so.

Don't you think people might get creeped out at this?

anon

Marvis beat me to an answer, boooo.

Eugene

ask anon

Peter Tamaroff

@anon Duel! Marvis is really busy these days,

anon

different strokes for different folks

Eugene

@anon watchoo talkin about willis!

Peter Tamaroff

02:38

@anon I'm gonna make him an offer he cannot refuse.

anon

@Eugene heh, that show stopped airing half a decade before I was born.

Eugene

i've never watched it either. i've watched avenue-q though

Peter Tamaroff

I loathe this question

anon

I think it's funny, and classic skull.

Peter Tamaroff

@anon classic skull?

anon

02:45

Saying something is "classic [person's name]" means it's characteristic or representative of them.

Peter Tamaroff

@anon Oh! skull**patrol**. Didn't pay attention to that.

My answer here is right, right?

Today I screwed some simple stuff, like writing $\frac{10}{1.86}=\frac{100}{186}$

anon

there might be an implicit interchange of limits in need of justification, but I don't feel like chugging through the details

Eugene

03:08

why is batman cheaper in the US than in canada

bah

anon

comic books, animated series, movies?

Eugene

animated series.

Dylan Moreland

04:17

I don't really understand the negative spacing here.

Frank Science

Let me test the Chat Rules.

Dylan Moreland

@FrankScience How? F-bombs?

Frank Science

about [title]

Frank Science

04:48

[title]math.stackexchange.com/questions/159752/…

[title]math.stackexchange.com/q/159752/23875

mixedmath

what are you using [title] to mean?

Frank Science

If you asked your question on the main site, please don't post it on the chat. It will get a lot of exposure without that. If you do choose to post it, title is a better format; it is compact and easier on the eyes.

In chat rules

title

I get it.

mixedmath

oh - now I see

you should actually write the title out, though

Frank Science

For example, My problem?

mixedmath

e.g. a question about such and such that happens to link to your question

yeah - like that

Frank Science

04:52

Some days ago somebody warned me that I should avoid pasting the URI to chatroom.

I found that the URI expanded to the contents automatically

Dylan Moreland

I don't think it's such a big deal, to be honest.

But others seem to disagree.

It's the same syntax (Markdown) for links as on the main site.

Frank Science

What do you mean by "a big deal"?

mixedmath

the only point is that we don't want needlessly duplicated work, right? So some people answering on the main, while others more or less write out the exact same here; and there's a reputation discrepancy

but I don't really care too much either

Dylan Moreland

@mixedmath I think the FAQ refers to the fact that, say, the following link

2

Q: Proof about $z\cot z=1-2\sum_{k\ge1}z^2/(k^2\pi^2-z^2)$

In Concrete Mathematics, it is said that $$z\cot z=1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}\tag1$$ and proved in EXERCISE 6.73 $$z\cot z=\frac z{2^n}\cot\frac z{2^n}-\frac z{2^n}\tan\frac z{2^n}+\sum_{k=1}^{2^{n-1}-1}\frac z{2^n}\left(\cot\frac{z+k\pi}{2^n}+\cot\frac{z-k\pi}{2^n}\right)$$ The trig...

is really huge and disrupts the flow of chat.

Frank Science

@DylanMoreland When MathJax works, it is huge.

mixedmath

05:04

ah, I suppose someone might care about that

especially on small screens

Dylan Moreland

I wonder whether there's been a meta SO thread about it.

Frank Science

The thread in meta math SE

Dylan Moreland

@FrankScience I wasn't clear: a thread in which someone suggests that the chat links are way too large.

Frank Science

What?

Dylan Moreland

Oh, never mind.

Frank Science

05:16

Incidentally, should we avoid syntax errors when we're chatting?

Benjamin Lim

@DylanMoreland hey

I used the urysohn lemma in my analysis exam

not sure if I used it right!!

Frank Science

Take-home-exam?

Benjamin Lim

no

final analysis exam

Frank Science

mid-term exam is take-home?

Benjamin Lim

05:41

no

it is not take home

Frank Science

I find that the exam of Concrete Mathematics in Stanford is take-home, so I don't know whether take-home exame is in fashion.

mlbaker

06:41

haha, hey

Frank Science

07:57

$\iff$

What's difference between $\Leftrightarrow$ and $iff$?

anon

one's bigger than the other

Frank Science

$\Longleftrightarrow$ and $\iff$.

anon

apparently nothing

David Wallace

@RagibZaman If you can't be certain, then why would you make such a serious allegation?

anon

actually I think \iff has extra space on the outsides of the actual symbol

Frank Science

08:00

Let me try: $A\iff B$, and $A\Longleftrightarrow B$.

Yeah

robjohn

10:54

a pin drops

anon

ouch

robjohn

@anon Sorry, did you find a pin?

anon

wut

robjohn

@anon you seem to be uncomfortable, and I seem to have dropped a pin. I thought there might be a connection.

a connection between the pin and you :-D

anon

Well obviously I was saying "ouch" because there was a sudden loud noise in the vicinity that hurt my eardrums.

robjohn

10:58

@anon Oh, sorry. I will drop my pins more lightly in the future.

That's odd, I didn't see the Laplace Transform answer to this question until just now, but it says it was posted 5 minutes before mine. Weird.

anon

I actually didn't see yours until long after the LT one. (In fact I pointed out to Tenali there was a typographical issue in a now deleted comment.)

robjohn

@JonasTeuwen: Howdy! have you returned home from Spain?

@anon I am just getting blind in my old age :-)

anon

11:14

Any idea why if G is abelian and G/H is free abelian, H is a direct summand?

robjohn

@Didier: good day! haven't seen you here in a while.

anon

11:44

Hmm: Given a transversal $X=\{\ell_i \}$ of the coset space $G/H$ we can define a function $f:X\to G:\ell_i \mapsto \ell_i$ and extend it via universal property to a homomorphism $\bar{f}:G/H\to G$. Then, because $G$ is abelian, $\bar{f}\cdot\mathrm{Id}:G/H\times H\to G$ is well-defined and also surjective... this proves the situation for finite groups, but not sure how to extend it. Going to bed anyway.

robjohn

@anon Good night. May the answer come to you in a dream :-)

anon

(Unless there's a simple way to see the kernel is trivial..)

Oh, the kernel has to be trivial, because $\ell_i^n h=e\implies (\ell_i H)^n=H$ contradicts freeness of $G/H$! I can rest easy now.

Chris

12:09

Hi.

robjohn

@Chris Hey there :-)

Chris

What did it happen with user9413?

It was a teacher from India if i remember well, Chandelasar or something like that.

robjohn

@Chris I don't know who that is?

Chris

He provided with a nice anwer for one of my integrals, but now i saw that it appeared "user9413" instead of his nickname.

He had a gravatar with John Lennon.

robjohn

@Chris ah, so that means that they changed their nickname

Chris

12:12

I see.

I hope it's not a removed user...

robjohn

@Chris which answer is that?

Chris

12

Q: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$

I need to compute the following integral: $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$$ At the moment i don't figure out any easy way to follow. Any support here is very welcome. Thanks.

real-analysis integral definite-integral

@robjohn: It's an elementary approach, a thing that i appreciate a lot. Well, as you posted as a comment a while ago, i like to see a lot of solutions!

robjohn

@Chris The link to his profile is dead. Perhaps he did leave.

Chris

It's Chandrasehkar.

@robjohn: maybe..

Ilya

13:30

wow, 5 people in the row

Benjamin Lim

14:28

hey @Ilya

dosvidaniya

robjohn

@Ilya Hey there, stranger!

Benjamin Lim

@robjohn hey :D

What do you think of munkres' analysis on manifolds?

I want to do a reading course on that next semester

robjohn

@BenjaminLim G'day, mate ;-)

@BenjaminLim I have never read it, but I would like to get or read a copy at some point.

Benjamin Lim

ok

my several variable calc is shit

because

and I would like to see how some of that interacts with like mnanifolds in R^

robjohn

@BenjaminLim I never got a chance to look at your question. If it is something you are still interested in, I will look now.

Benjamin Lim

14:30

@robjohn which problem?

robjohn

@BenjaminLim the notation question

Benjamin Lim

@robjohn ah it's ok now!

I got that

thanks anyway :D :D

geometry/manifolds is something I have not seen a lot of

it's all been like always algebra algebra algebra

robjohn

@BenjaminLim analysis has not been high on your list?

Benjamin Lim

@robjohn no not really

but like I said I would like to do a course with like multivariable calculus + manifolds

tomorrow I am meeting my supervisor

robjohn

@BenjaminLim Is meeting your supervisor stressful?

Benjamin Lim

14:35

what is?

to discuss subjects for next semester

and sometimes it can be :D

@robjohn Next semester subjects line up:

algebraic topology and linear groups

robjohn

@BenjaminLim what's a linear group?

Benjamin Lim

GL_n

PGL_n

robjohn

Ah

Benjamin Lim

those guys

robjohn

okay

Benjamin Lim

14:38

my several variable analysis is not good

i'm looking at munkres

and it seems ok

Sl_n

robjohn

@BenjaminLim It is nice to find a book on a subject that you are not too familiar with that you can understand.

Benjamin Lim

yes

exactly

that's why I also had a look at loring tu's introduction to manifolds

and john lee's smooth manifolds

robjohn

14:54

@BenjaminLim I have mainly been raised on Rudin and later on Stein, so I am older school than those, I think.

Benjamin Lim

hmmm

you mean like elias stein harmonic analysis?

from what i've been told

rudin gets not so good in several variables

robjohn

@BenjaminLim he was my advisor

Benjamin Lim

stein?

for phd?

robjohn

@BenjaminLim I read Spivak on several variables.

Benjamin Lim

ah ok

you mean like calc on manifolds?

robjohn

14:55

@BenjaminLim pretty small book, but it is good.

Benjamin Lim

yes

however some say it's not as advanced as munkres

robjohn

@BenjaminLim that's probably true.

Benjamin Lim

yes

i am meeting him in exactly 9 hours

robjohn

@BenjaminLim Do you need to prepare something for the meeting?

Benjamin Lim

not really

it's not anything technical

like having to do actual maths and stuff

just sit down and discuss what to do next, where to head in my life etc

he's a complex analyst

his homepage is there

Benjamin Lim, Canberra, Australia

6.8k 2 16 35

linear-algebra commutative-algebra homework abstract-algebra field-theory ring-theory real-analysis

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