Mathematics - 2016-12-02 (page 10 of 11)

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7:01 PM

@Astyx a serious hint is also the points for the excercise. 5 of 20 for the whole supexcercise. 15 for the other. so one might guess it's really easy, but not me appearantly :/

Indeed

I thought of a better to read notation for sums

a box around the to sum term. and the limits above and below

?

user image

dunno, just a thought ;)

$\Sigma$ works perfectly fine, doesn't it ?

Random Variable

7:07 PM

@DanielFischer Then we need to justify that $$\oint \int_{0}^{\infty}\frac{e^{-a\sqrt{1+x^{2}}}}{\sqrt{1+x^{2}}} \, J_{1}(bx) \, dx \, da = \int_{0}^{\infty} \oint \frac{e^{-a\sqrt{1+x^{2}}}}{\sqrt{1+x^{2}}} \, J_{1}(bx) \, da \, dx $$ for $\text{Re}(a) >0$, right?

Nested summations would be awkward

with $\Sigma$ it all looks way bigger than it is. also it is "sometimes" not very clear where it ends. with the box those problems would be solved.

What do you mean ?

Yes. But the integrability of $\dfrac{1}{\sqrt{1+x^2}}J_1(bx)$ makes that trivial.

$$\sum_{k=1}^n k$$ is as clear as your notation

7:09 PM

@Null This get really troublesome really quick, for example $$\sum i + i^2 + \left( \frac{3}{i} \right)$$

Then imagine also summing over $j$ after that, or even a triple sum

@Krijn ah so, triple boxes would basicly turn my idea to shreds hehe

(because then its not that better)

Also the fact that $\sum$ without any bound is used for series, and $\sum_i$ by physicists to avoid writing the bounds makes sense

And that wouldbe difficult to reproduce with your proposition

Physicists skip the sum directly

Not all of them

Lazy physicists ...

the thing I learned today: not all limit proofs end with $<\varepsilon$. The rest is nil.

7:15 PM

?

@Astyx pastebin.com/S8sJ813c

i only saw proofs that use alot of bounds by above

arriving at epsilon

Random Variable

@DanielFischer Sometimes you get so used to justifying something in a particular way that you forget that there are other ways. But what about arguing that the formula holds for $\text{Re}(a) =0$? Is that where the continuity argument comes in?

Does anyone here know about automata theory ?

Hey-men-whatsup

7:34 PM

Need book recomendation on mathematics or calculus that doesn't give plain(basic) examples..., I've already wasted some money

What field of maths ?

How do I proceed? pastebin.com/VfbWpb2c

@DanielF, bonsoir, despite your snide remarks about us cogniscenti who insist $0\notin\Bbb N$.

@Null What's the problem ?

Do you know operations on limits of sequences ?

@Astyx i want to avoid the leading coefficient law. because we did not have that yet

Hey-men-whatsup

7:38 PM

@Astyx, I bought this book but the examples are so plain, so I'm not sure wether I understand the material after reading the books for hours

amazon.com/Calculus-Early-Transcendentals-Jon-Rogawski/dp/…

@Null Well the numerator tends to $3$, the denominator to $2$, thus the whole to $3\over2$

@RandomVariable Yes, that's just the continuity.

@Hey-men-whatsup: What do you mean by "examples are so plain"? That's one of the better calculus books. If you're complaining that there is not an example for every sort of homework problem, that's appropriate.

@Astyx to not sound handwavey i'd have to show that c/n tends towards 0, the same for c/n^2 or? (c some constant)

@TedShifrin There is no shame in being wrong :)

7:39 PM

@Null: Do the estimates with triangle inequality (done appropriately) for numerator and denominator.

@TedShifrin The truly enlightened know that sometimes $0 \in \mathbb{N}$, and sometimes not. Whichever is more convenient.

How can I prove rigorously that $\forall x\in \mathbb R : 0 \le x^{\frac{1}{x}} \le e$ ?

No, @DanielF, the truly enlightened have a notation like $\bar{\Bbb N}$ so there's no confusion :D

You need calculus, @Mahmoud.

@Mahmoud $x^{1/x} = e^{\ln x\over x}$

@Astyx: Nor shame in ignoring snide Frenchmen :P

7:41 PM

@Astyx Wait .. Why ?

@TedShifrin That looks a bit like $\mathbb{N}\cup \{\infty\}$ to my topologist eye.

In France we use $\Bbb N^*$ to say $\Bbb N - \{0\}$

@TedShifrin I don't think calculus is a big problem ..

@DanielF: I've spent my life with $\Bbb N$ starting with $1$. I'm not going to change.

Because $x = e^{\ln x}$ @Mahmoud

7:42 PM

@TedShifrin No need to. As long as we agree that compact spaces are Hausdorff by definition.

No, I don't agree on that one, either.

Heretic.

Fool.

:P

$\mathbb{N}^{\times}$ is used to express: "discard elements, so my stuff makes sense"

Don't know that notation

7:43 PM

$R^\times$ denotes units in a ring $R$, @Astyx :P

Hey-men-whatsup

@TedShifrin, the examples on the book are too simple, I need variations on the examples

Yes, with a $\times$, mine is a $*$

Hey-men-whatsup

so I can learn better

@Astyx Couldn't we choose a different base for the log, say $log_{10}$ ?

@Hey-men-whatsup: Unless you mean you want the book to be a thousand pages longer with an example of every sort of problem, I don't understand. The examples in that book are varied.

Hey-men-whatsup

7:45 PM

@TedShifrin , I mean like this

You learn by working problems, @Hey-men-whatsup, and not just reading what someone else does.

@DanielFIscher What studies did you do to become topologist if I may ask ?

Hey-men-whatsup

user image

@Mahmoud well $x = 10^{\log_10 x}$

a rigorous definintion of the asterix(or cross) at the top of a field would be sweet

Hey-men-whatsup

7:46 PM

the book is the "in class"

LOL @Hey-men-whatsup

@Hey-men The car is obviously red

@Astyx I'm not a topologist. One of my eyes is.

Ah my bad :)

Random Variable

@DanielFischer Perhaps we're saying the same thing, but I was thinking that because $\frac{1}{\sqrt{1+x^{2}}} \, J_{1}(bx) $ is Lebesgue-integrable, the DCT could be use to justify that $$\lim_{\alpha \to 0^{+}} \int_{0}^{\infty} \frac{e^{-(\alpha + i \beta)\sqrt{1+x^{2}}}}{\sqrt{1+x^{2}}} \, J_{1}(bx) \, dx = \int_{0}^{\infty} \frac{e^{-i \beta\sqrt{1+x^{2}}}}{\sqrt{1+x^{2}}} \, J_{1}(bx) \, dx$$

7:47 PM

What studies did one of your eyes do then ?

@Hey-men-whatsup: Let's be specific. What problem are you trying to do and why are you stuck?

Hey-men-whatsup

I've wasted money buying some books lol

There are far worse books than that one.

@RandomVariable Yup.

No book will have an example of every sort of problem you want. And I doubt you want a really hard math book that deals mostly with proofs.

LOL @Astyx

@DanielF might need to borrow six or seven of my eyes.

7:48 PM

What does Hausdorf really mean ? I've heard it a lot on this chat. Is it equivalent to "separated" ?

@Astyx Yes, but Hausdorff is cooler.

It depends on what "separated" means.

$T_2$

I never learned all the $T_i$.

I mean, $T_{3.5}$ and $T_{4.5}$? Seriously?

I don't know about $T_{4.5}$, only $T_{2.5}$ and $T_{3.5}$ as non-integer indices.

7:50 PM

@Hey-men-whatsup: You're frustration is exactly where you must choose two paths, weather to be a mathematician or not. Either you choose to go out and explore your inquisitive nature (which more than likely you won't be able to consult a text on because it is your thought process) or you only answer questions which are being asked of you. That's the determining nature of a mathematician. Books are intended to get you started, not tell you everything.

"separated" means there exists disjoint neighborhoods of any two distinct points (if I'm not mistaken)

@Faraad: I believe @Hey-men-whatsup is an engineer. He really doesn't want to be a mathematician. He wants to see more complicated applied problems worked out. Am I correct?

@Astyx: Then it's Hausdorff :P

Right :)

@Astyx: M'enfin, je m'en fous :)

BTW, hi @Faraad.

"M'enfin" :D

Hey-men-whatsup

7:52 PM

@Ted

user image

@Astyx Hausdorff means that for any two point x,y in your space, you can separate them by disjoint open sets. The open sets are define by your topology T i.e Hausdorff is an additional requirement on T.

Hey-men-whatsup

I mean you got me

not extreme math, but more like pratical

ignorants, ignorants everywhere lol

Seriously, to do what you ask would make the book 1000 pages longer.

But let's discuss a specific question.

Hey-men-whatsup

professor in the first picture, keeps laughing at me

help me..

7:54 PM

Every textbook I've written, many students complain that there aren't enough easy examples. Or that I haven't worked every homework problem for them in the book. Mathematics and engineering just require thinking.

Easy exercises are easily found on the internet anyways

Well, if you post all your homework on MSE, it'll all get done sooner or later by someone. No learning ... but ...

@TedShifrin: It doesn't matter. That's the problem. Just because he is an engineer, that shouldn't mean he shouldn't have a good understanding of what mathematics really is. With this understanding he would easily realize the intent or purpose of a text on mathematics. I really don't like books which divide mathematics, in the sense that they are titled, mathematics for engineers, mathematics for the liberal arts major...It is all under the same umbrella.

@Faraad: My point is that he's interested in learning to solve challenging problems, not in being trained to do proofs. I'm fine with that.

Hey-men-whatsup

@FaraadArmwood, well "time" will against that, I've got no choice

but rapid understanding

7:57 PM

@FaraadArmwood I don't want my house being build by a person who did only learn to copy houses.

Solving engineering problems only requires practice, basing on my humble experience

@Null: Exactly my point. I see you get what I'm saying.

I'm just arguing that to understand calculus to do hard problems does not require that you know subtle proofs of theorems.

Lots of practice

Of course, everyone should truly know why the Fundamental Theorems of Calculus work. But the ultimate generality with epsilons? Not needed.

European style still emphasizes proofs more than the US. But the US has watered down the courses and doesn't even expect medium-level computational competence.

7:59 PM

What is the difference between analysis and calculus ?

It is easy: if you only ever try to solve already solved problems or variations of them you will never be able to solve a true problem.

@Hey-men-whatsup: Again, I understand you completely, but you have way more time than you think. Yes, it is always a struggle balancing theory and application; we should be able to do both.

<--- withdraws from the conversation.

Hey-men-whatsup

@FaraadArmwood, correct.

@Astyx i find it funny that we talk about 0 after all those days. you are quite sticky hehe

8:01 PM

I was not the one to start talking about it this time :p

But I am still shocked that for some people, $(\Bbb N, +)$ is not a monoid

5

Hey-men-whatsup

user image

let's do it then.

@Null and to answer your previous question, this depends on what you call handwavey :p

@Astyx: And why should I care whether $\Bbb N$ is or is not a monoid?

@Astyx well, my corrector wrote: "this is not lectured yet, please only use the content of the seminar"

math is tasteless without rigorous proofs

8:05 PM

It seems very useful in automata theory for instance

(the last time i tried to justify a limit)

@Astyx: Well, that's far from anything I would ever have used or thought about. Next?

Not true, @LeGrand ...

so i pull the hammer and simply proof every single step that is not clear @Astyx

@Ted On another note, would you be interested in answering this question?

@DanielF: This is actually not something I've ever sat down and worked out. The octonions are wonderful things, but I've never played with them.

8:08 PM

@TedShifrin I saw a fractal image of the Mandelbrot set and I want to work it out for myself, may you please tell me what background do I need to understand this ?

complex numbers @Mahmoud

Heeyy yooo Teddie

And limits

@Mahmoud: Yeah, it's just about repeatedly evaluating a complex quadratic polynomial, but there's no way to "work it out" without a computer.

Howdy @Danu

I'm doing OK

8:09 PM

@Ted or a lot of time

I guess my thesis will probably be on this curvature-vs-characteristic numbers thing

@Astyx Is that really necessary? I feel like the idea behind the fractal image of the mandlebrot set is pretty far removed from the actual construction of it.

@Astyx: Really?

The rigorous mathematical foundation of fractals is pretty subtle.

Ok thanks $:)$ But @TedShifrin But ''by computer'' you mean the computations ?

Hey-men-whatsup

@TedShifrin, are you the writer of this book? Cool...

amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/…

8:10 PM

You're really just testing how fast a function converges and assigning a number to it.

Yes, @Mahmoud ... the picture you see is colored according to where points go in the limit.

@Hey-men-whatsup: Yeah, that's one of 'em.

@Danu: That sounded interesting to me.

@TedShifrin I always wondered... Is this stuff somehow relevant for mainstream math?

Hey-men-whatsup

that looks advanced, right?

@TedShifrin Let's hope so! I guess I'll be reading the papers Kotschick references now

Hi @Danu, how are you ?

8:11 PM

I'm excited to see if I can make sense out of Gromov & Lawson's paper

@Hey-men-whatsup That looks very much like Ted's book.

Chaotic dynamical systems are important in a lot of parts of math, @Danu. Not in stuff I think about, but ...

are there any solutions in integers to $x^n-2y^n=1$ for $n>2$? I have only managed to prove that for each $n$ there are finitely many solutions

@Astyx I'm good---how about you?

@TedShifrin Mhm... I like it when things are as little chaotic as possible

Though I guess symplectic geometry is somehow related to dynamics somethingsomething

@Hey-men-whatsup: It covers stuff in the last third of your textbook and beyond, using linear algebra and largely proof-based.

8:12 PM

I'm fine thanks :)

LOL, @Danu: As an ex-physicist, you shouldn't be mumbling "somethingsomething."

@TedShifrin I already argued that physicists don't care about symplectic geometry a few days back :P

Well, any physicist who plays billiards might care :P

Hahaha

Just like they should care about orbifolds? ;)

@Danu Would you have time now for a short introduction to Lie algebrae ? Or is now not the right time ? :)

8:14 PM

Actually, orbifolds probably are more important ... Since there are finite symmetry situations in physics not just occasionally.

Hi.

@TedShifrin I know :) String theorists care ;)

I guess I'm ATCing in the winter.

Hi Mike

@MikeM. G'night. Would you have a reference for this guy? Daniel just bugged me about it.

8:14 PM

@Astyx I'm really not the right gguy to teach you about Lie algebras. I just know a little bit about what physicists care about.

What is ATC?

@Danu ah alright, no problem :)

What specifically are you interesting in knowing, @Astyx?

Advance to candidacy.

Oh, sure. Of course you should.

8:16 PM

@Ted: Doesn't it just seem like they precimpose with the inverse?

We proved this theorem by Gompf on symplectic 4-manifolds with prescribed (finitely presentable) fundamental groups @Mike :)

So the action is just the right action. $xg^{-1}$.

I was very excited by the corollaries :P

@MikeM: To be honest, I didn't read it through carefully.

@Ted Just a basic, short introduction : what it is,why it's interresting, and what applications it has

8:16 PM

I certainly don't have a reference. It's too specific to expect one.

@Danu It'd a nice theorem.

Lie algebras as opposed to Lie groups? @Astyx ... Vector fields on a manifold form a Lie algebra (an infinite-dimensional one). So do certain vector spaces of matrices.

good afternoon everyone

@MikeM: I wouldn't be surprised if there were something relating to this in Steenrod. Sadly, I didn't keep him.

Hi @meow

Hi, @meow. They let you out of school already?

8:18 PM

@TedShifrin students are dismissed at 2:40

Ah.

@Ted That might be a little bit too short for me to understand .. ^^

@MikeMiller Right :) And it turned out that my supervisor also proved something related to it (essentially by putting together some theorems from gauge theory) :)

Well, @Astyx, it's hard to give a 10-minute lecture in here without some focus. Why do you say Lie algebras instead of Lie groups?

@Ted I don't know

That's the term I've heard before I guess

8:20 PM

I'm really excited to learn about how gauge theory restricts topology of manifolds

Lie algebras are in some sense easier because they're vector spaces. But they arise geometrically because they are the tangent space (at the identity element) of a Lie group.

@Danu: You're following in Mike's footsteps. I hope you don't trip :P

@TedShifrin We'll see... I'm not sure I'll have much of a chance to learn a lot about gauge theory anytime soon... I'll have my two courses but I will otherwise will probably be busy for the next 9 months :\

@Astyx: Abstractly, a lie algebra g is a vector space who's binary operation is denoted [,] i.e the Lie bracket, which has certain properties. There are about 3-4 useful, equivalent definitions of what a Lie algebra is. Ted, gave you are very good one, which is the geometric one.

By the way, I was looking at a cool workshop: wwwf.imperial.ac.uk/~at515/bigworkshop.html

Well, @Danu, I assume you're still happy you changed course ...

8:23 PM

It's sorta 100% student-presented, I think. Maybe I could even give a talk...

@Astyx can you look over excercise 4(a)? docdro.id/8SALcFx

@TedShifrin I'm very happy doing math :) Still feeling very inept though ;)

I'm still inept.

@Faraad And why are these structures interresting ?

@Null Sure

@TedShifrin You've earned it

8:24 PM

LOL ... ponders

@Astyx: That question, opens a huge can of worms. Studying the Lie algebra of a Lie group in a lot of situations tells you what the Lie group must look like. My co-advisor Mike Cohen has notes up for the course that he is teaching this semester (Lie Groups and Lie Algebras).

@Null I have no idea what your proof is about exercise 1.b

@Astyx no prob, just skip to 4 (a)^^ its the fraction stuff, 1/n, as n to infinity

@FaraadArmwood Thanks, I'll look that up !

@Null Seems fine to me

@Astyx thanks. just wanted to nail that c/n part ;)

8:28 PM

@Null but I find exercise 1 is very poorly written :p

@Astyx 1(a) was a calculating task. And 1(b) i frankly just copied and changed it to my needs. so surely it looks wierd :D

It also looks wrong

That typo ..

haha^^

Even my computer says I'm ugly ...

:p

@Astyx only wrong, whatever that means to your computer :D

8:32 PM

Wrong seems bad enough

I just want to check real quick: if im working in the $\Bbb R$ then $<x,y>=<\overline{y,x}>=<y,x>$ correct?

If $x$ and $y$ are real vectors, then, yes, the usual hermitian inner product is symmetric.

perfect, thanks!

Use \langle \rangle, not < and >

^

8:35 PM

@Danu: We can't all be LaTeX perfectionists.

But we should strive

Those of us who type up hundreds of pages of notes or books should be, of course.

i knew that existed but couldnt remember and got lazy :)

You're not a true perfectionist until you expect others to be perfectionists :p

@TedShifrin I was thinking I could be a math typesetter if not a mathematician :P

8:36 PM

I typeset all my books, @Danu, actually, so I was semi-seriously thinking of doing this when I retired. But I decided to be a bum instead.

@Astyx i expect others to be perfectionists and myself is lazy haha

@TedShifrin I think I'd like it. But where to find such a job?

There are companies in the US that literally do typesetting for publishers. You could inquire with Springer how they do this ...

@Astyx i found a solution to the problem

@meow Which problem ?

8:38 PM

@TedShifrin Hmm

Mathematician is still #1 haha

Good @Danu

@Astyx the 4-cycle problem

Oh, what is it, @meow?

I thought about it a little.

@TedShifrin: Is there a good enough demand for older texts to be updated i.e rewritten in tex?

Sorry, but I have no idea what you are talking about ...

8:40 PM

@Ted (2 3 4 5) (4 2 3 5) (4 3 5 1)

what would be the most general statement that includes the limit of $\frac{n^3}{2^n}$? $\frac{n^{k+1}}{k^n}$?

oops thats backwards

@meow What was the problem ?

@Faraad: There are copyright issues. You mean to make it freely available in .pdf form?

@TedShifrin: No, just to rewrite and resell.

8:41 PM

@FaraadArmwood haha

ok fixed

@Null: No, $n^k/2^n$, basically.

@Faraad: I doubt publishers want to invest in that.

That doesn't work, @meow. Oh, do I read left to right or right to left?

Happy Early Holidays!

@TedShifrin that's the sequence;

@TedShifrin youd apply $(1234)$ first then $(2345)$ then $(4235)$ etc. etc.

So right to left

8:43 PM

yeah :P

wait did i copy that right

one second

@Ted What is the problem ?

Then I get $(1\, 3\, 5)(2\, 4)$.

He's trying to write $(1\,2\,3\,4)$ as the product of $4$-cycles, no two of which fix the same things.

Is (1 2 3 4) not a solution ?

@Astyx theres a restriction

@SimpleArt This is mesmerizing

8:45 PM

@Astyx: Nontrivially.

@Astyx Tell me about it. I spent about 5 hours today messing around with it.

@SimpleArt Totally worth it

There should be a way to see this geometrically, @meow. Do you know that $S_4$ is the group of symmetries of the cube?

yes

It's even going in my bookmarks

8:46 PM

So, if you have a solution, we should see it geometrically :)

@TedShifrin: That's a shame.

@Faraad: I can't even get Wiley to do a new printing fixing all the typos I've found in the blue book. :( I even provide them all the LaTeX.

I'm pretty much done with traditional publishers.

I think @MikeM was at one point thinking he'd retype Gunning's various Riemann surfaces books in LaTeX, but he lost interest in it. Probably a good thing.

I have an earwig of our hous' bell. disgusting

@Astyx Now I wish they had let me do double inequalities while in polar mode...

@Null: Auf englisch geht das nicht :)

8:49 PM

@TedShifrin sorry, my english is not the yellow from the egg :-)

Ni das. :)

@Ted What does nontrivially imply here ?

He wants it as a product of at least $2$ $4$-cycles, but no repetition of cycles.

"Auf englisch" ?

Is my german that bad ?

Yes, I believe I'm correct.

I mean, it's been 43 years since I studied German, but ...

Il faut demander à @Null.

8:52 PM

@TedShifrin heres a solution

(3 2 5 4) (4 3 5 1) (3 5 6 7) (7 6 5 4) (7 4 5 3)

It must be because my german prof was absent today, this justifies my mistake

@TedShifrin: Well I think publishers view it from a more economical standpoint. I mean this to say, it is the case that trig, pre-calc, and calculus texts are required for almost every major so they would be willing to put work into texts as those. However, for more junior/senior level advanced mathematics, there usually is no strict requirement by the instructor. What I am asking is why don't authors make there text freely available after some time?

@meow: Before I check — my comment about Cube is wrong, since we need to go into $S_n$ for $n$ large. Do you believe you cannot do it with less than 7?

@Faraad: We sign contracts. The publishers own the books. They'd have to give or sell me the copyrights back.

@meow: It still is wrong. $1$ goes to $3$.

@TedShifrin you cant do it for n = 4 since that would require using a cycle congruent to (1234)

@TedShifrin what? i tested it

wolframalpha.com/input/…

Yeah, of course, @meow. But yours is still a mess. $2$ goes to $5$.

I dunno.

8:55 PM

@Astyx in english=auf englisch. But "in englisch" would be correct too!

We are in $\mathfrak S_4$ here right ?

@TedShifrin: Oh wow! I never knew that. That's cazy. Its very upsetting that they are so money hungry they are willing to deprive people of learning.

$\mathfrak S_n$, $n\ge 4$.

@Astyx or at least everyone would understand what you mean. therefore...

@Null Ah, right ! That's what I thought, thanks for the clarification

8:57 PM

@FaraadArmwood You're surprised at this? :P

@Faraad: I decided to leave the diff geo notes available for free. Of course, when I'm dead, there will be no more website.

@Astyx but "auf" has quite a few meanings. "auf dem Klo"=in the bathroom, "auf dem Dach"=on the roof...

@meow: That calculation says you get the inverse of $(1\,2\,3\,4)$ if we get the ordering right.

Yeah this I know, I just wasn't sure about it being used for languages

Hey-men-whatsup

why do I see math proof in philosophy???? I'm drunk I need sleep...

0

Q: I am stuck on how to prove the contradiction of R(b,a) can anybody help me?

Here are some well-known properties of dyadic (2-place) relations: ∀xR(x, x) (Reflexivity) ∀x¬R(x, x) (Irreflexivity) ∀x∀y(R(x, y) → R(y, x)) (Symmetry) ∀x∀y(R(x, y) → ¬R(y, x)) (Asymmetry) ∀x∀y∀z((R(x, y) ∧ R(y, z)) → R(x, z)) (Transitivity) ∀x∀y∀z((R(x, y) ∧ R(y, z)) → ¬R(x, z)) (Intransitivity...

8:59 PM

@Astyx: Maybe usage has changed since I studied German. But we always said "auf" in referring to a language.

Anyhow, lunchtime for me. Bubye.

@TedShifrin see ya :)

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