Mathematics - 2015-11-25 (page 2 of 3)

6:01 PM

@robjohn It seems the value of the multivariable integral tends to $1$ while the number of variables tend to $\infty$, strictly increasing from $v\ge2$.

Agawa001

@AinzOoalGoal hey mr le pegase

robjohn

@Chris'ssistheartist my laptop is still chugging on the negative exponent 4 variable computation.

Chris's sis the artist

@robjohn different values all over.

Ted Shifrin

@robjohn: Hope you don't burn up your laptop the way AinzOoalGoal did.

robjohn

@TedShifrin how did they do that?

Ted Shifrin

6:03 PM

Ask him .... I really don't know.

Agawa001

i would smash it with a jackhammer if he wouldnt respond after an hour or so

Ted Shifrin

Salut, @Agawa.

Agawa001

salut , ça va bien bien Mr ted ?

Ted Shifrin

Oui, bien, merci, et toi?

Agawa001

On fait rouler Mr ted

Ainz Ooal Goal

6:18 PM

@Agawa001 hey

Ted Shifrin

What are you studying these days, M le Méchant?

Ainz Ooal Goal

@TedShifrin Probabilities

Ted Shifrin

Unlikely circumstance, but ok :)

Chris's sis the artist

I also tried to see that integral using differential equations, that is

$$y''+y=\Gamma(1+x)$$

robjohn

@Chris'ssistheartist Inverting that differential operator is not a difficult process

Chris's sis the artist

6:33 PM

I considered the case $a=1$.

robjohn

@Chris'ssistheartist $a$?

Chris's sis the artist

2

Q: Evaluating $\int_0^{\infty} \frac {e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$

I am trying to evaluate this integral: $$\int_0^{\infty} \frac {e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$$ for $a \in \mathbb R$. (of course with $a > 0$) Any ideas?

robjohn

Ah

Chris's sis the artist

I considered $$y(s)= \int_0^{\infty} \frac {x^s e^{-x}}{a^2 + \log^2 x}\, \mathrm d x$$ where above I chose $a=1$.

robjohn

@Chris'ssistheartist Yeah, I guess that from the differential equation

Chris's sis the artist

6:35 PM

@robjohn It's a nice differential equation. I wonder if one can get a nice form. My Mathematica isn't able to do it.

robjohn

@Chris'ssistheartist It is easy to invert if you can integrate the function against $e^{ax}$ for complex $a$.

Chris's sis the artist

@robjohn Maybe I should consider the differential equation while taking the limit from $s$ to $0$, and hope not to say a stupidity.

I'm interested in that function near $0$.

user1667423

How can I show that P(B) = 1 entails P(A) = P(A ^ B) from the probability axioms?

Chris's sis the artist

@robjohn maybe you can do it with some complex analysis. It looks like those questions that ask for complex analysis.

robjohn

6:57 PM

@Chris'ssistheartist I don't know. I may try it later.

@Chris'ssistheartist @r9m Doh! of course it works for both of those integrals... for any real $a$, we have $$\int_0^1\int_0^1(xy)^{axy}\,\mathrm{d}x\,\mathrm{d}y =\int_0^1x^{ax}\,\mathrm{d}x$$

that includes $a=\pm1$

Chris's sis the artist

@robjohn Nice.

robjohn

@Chris'ssistheartist $a=0$ requires a division by $0$ in my proof, but that case is trivial.

Chris's sis the artist

@robjohn You might ask people to prove that $$\int _0^1\int _0^1\frac{1-(x y)^{x y}}{x y \log (x y)}\ dx \ dy=\int_0^1 \frac{1-x^x}{x \log (x)} \, dx$$

@robjohn You might propose something like that to a magazine.

The one above looks gorgeous.

It looks nicer after putting 1 in front.

robjohn

@Chris'ssistheartist As long as the integrand is a function of $x\log(x)$.

Chris's sis the artist

@robjohn I integrated from 0 to 1 with respect to a.

robjohn

7:04 PM

@Chris'ssistheartist did it work?

Chris's sis the artist

@robjohn Sure! Numerically is perfect.

robjohn

@Chris'ssistheartist which integral are you talking about?

Chris's sis the artist

@robjohn $$\int_0^1\int_0^1(xy)^{axy}\,\mathrm{d}x\,\mathrm{d}y =\int_0^1x^{ax}\,\mathrm{d}x$$

@robjohn integrate both sides with respect to $a$, from $0$ to $1$.

robjohn

@Chris'ssistheartist Oh, yeah. It is true for all real $a$

Chris's sis the artist

@robjohn Many other integrals can be got from exploiting that result. It's very nice, indeed.

@robjohn Just look at this one ...

$$\int _0^1\int _0^1e^{\large (x y)^{x y}}dydx=\int_0^1 e^{\large x^x} \, dx$$

robjohn

7:09 PM

@Chris'ssistheartist If $f(x)=g(x\log(x))$, then $$\int_0^1\int_0^1f(xy)\,\mathrm{d}x\,\mathrm{d}y =\int_0^1f(x)\,\mathrm{d}x$$

Chris's sis the artist

@robjohn Yeap.

@robjohn above I replaced a by k, then I divided both sides by $k!$ and performed the summation from $k=0$ to $\infty$.

robjohn

@Chris'ssistheartist yeah, but $e^{x^x}=e^{e^{x\log(x)}}$ so it fits my criterion

Chris's sis the artist

@robjohn If I'm not wrong this relation can be used as a recurrence relation to get multiple integrals, long crazy integrals.

Just treat the inner integral in the left side considering the initial relation, that is $\int_0^1\int_0^1f(xy)\,\mathrm{d}x\,\mathrm{d}y =\int_0^1f(x)\,\mathrm{d}x$.

Something like $$\int_0^1\int_0^1f(xyz)\,\mathrm{d}x\,\mathrm{d}z =\int_0^1f(x y)\,\mathrm{d}x$$

and then the process can continue indefinitely

robjohn

@Chris'ssistheartist if $f(x)$ meets my criterion, it is not necessary that $f(ax)$ does. Otherwise, your equality above would give the three variable equation.

Chris's sis the artist

@robjohn Yes, I see that.

@robjohn That works using the condition you stated above, of course.

robjohn

7:22 PM

@Chris'ssistheartist No, if $f(x)=g(x\log(x))$ then it is not necessary that $f(ax)=h(x\log(x))$

Chris's sis the artist

@robjohn Right. I only referred to the way I got my relation above with $f(x y z)$.

Owatch

Hello hello.

Chris's sis the artist

@robjohn Yes, right, it doesn't work always.

Owatch

So, uh, given function $f(x) = \frac{1}{xln(x)ln(ln(x))}$, I'm trying to find a concise way to list the domain.

I know it can't be negative, or zero, or $e$.

or $1$.

robjohn

@Owatch to which thread is that directed?

Owatch

7:30 PM

But I can't say it's $(1, +\infty)$.

What do you mean Robojohn?

It's not directed at any particular thread. It's a question.

robjohn

@Owatch to whom are you talking? (this is why I like using the arrows showing to what you are replying)

@Owatch Oh... I see.

Owatch

@robjohn To everyone. I don't like to ping since it sometimes irritates people. But I'll d it to reply.

robjohn

@Owatch it would need to be $x\gt 1$

If $x\lt 1$, then $\log(\log(x))$ would be imaginary

@Owatch or are you allowing complex values?

Owatch

If $x$ is $e$, then $ln(e)$ is 1, which is also imaginary.

Well, it's not, but because it's enclosed within another $ln()$, it becomes so.

@robjohn No, I am not allowing them.

robjohn

@Owatch then you would want $x\gt1$

Owatch

7:36 PM

$e > 1$, and is forbidden. Are you sure that suffices?

robjohn

@Owatch I am just looking at $\log(\log(x))$. That is real if $x\gt1$

Since it is in the denominator, $x\ne e$

Owatch

Why can't x = $e$, if it is in the denominator?

It doesn't work for this function, but you said the domain is $x > 1$.

which I thought $e$ would fall into.

user174558

@Owatch Your LaTeX is terrible, should be $x=e$, lol.

Chris's sis the artist

@JasperLoy How is it going today?

Owatch

You just waited till my 3 minutes were up, then told me -_-. I could have edited it.

user174558

7:43 PM

@TedShifrin I am sorry if I upset you, but I wasn't really specifically referring to you that day. Happy Thanksgiving. You are a great mathematician.

user174558

@Chris'ssistheartist So so. I have yet to look through my book.

user174558

@Owatch Owatch sounds familiar, I have seen you before in here...

anon

@DanRust @TedShifrin Let $w$ be the matrix with 1s on the diagonal, 1 in the upper right corner, and 0s elsewhere. Then $H_3(\Bbb Z)\times\frac{1}{n}\Bbb Z$ mod $\langle (w,-1)\rangle$ is torsionfree, has finite presentation $$\left\langle w,x,y,z~\left|~\begin{array}{c} [x,y]=z=w^2 \\ [x,z],[y,z],
\\ [w,x],[w,y],[w,z] \end{array}\right.\right\rangle$$ and abelianization $\Bbb Z^2\times\Bbb Z/n\Bbb Z$.

r9m

@Chris'ssistheartist really nice idea! :D .. (I still haven't computed the infinite series you showed me in chat the the other day ... I am lazy (level: Sloth ) :P)

Owatch

@JasperLoy You know it.

You have.

user174558

7:46 PM

@r9m I see your new picture.

user174558

@Owatch Yes, you are The Oracle or something.

r9m

@JasperLoy :-) One-Punch man (anime)

Owatch

Wait, what

Chris's sis the artist

@r9m @robjohn still there is a thing that bothers me, that OP asks teh following question that is a particular case of this problem

19

Q: Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$

I came across this nice identity: $$\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$$ Is there an elementary proof?

calculus definite-integrals gamma-function

On wikipedia I saw that integral in the left-hand side, not sure where, and it doesn't have a known closed form. Now, to find the generalization for that integral in the right-hand side, given this context, it seem like solving an open problem.

anon

@DanRust @TedShifrin I meant =w^n in the presentation.

robjohn

7:49 PM

@Chris'ssistheartist so that bodes ill for the other integral :-)

Chris's sis the artist

@robjohn lol, it seem so.

The constant given by that integral even has a name ... (let me remember)

r9m

@Chris'ssistheartist !! Nice Identity !! .. seems OP is trying to reach a generalization of some kind!

Owatch

So, are we all sure that the domain is actually $x > 1$ ?

I guess it works. !

r9m

@Chris'ssistheartist Nice there is a link in the comment under the question!

Owatch

But I thought $e$ would be required in it.

Chris's sis the artist

7:52 PM

Fransén–Robinson constant

Fransén–Robinson constant

The Fransén–Robinson constant, sometimes denoted F, is the mathematical constant that represents the area between the graph of the reciprocal Gamma function, 1/Γ(x), and the positive x axis. That is, The Fransén–Robinson constant has numerical value F = 2.8077702420285... (sequence A058655 in OEIS), with the continued fraction representation [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, ...] (sequence A046943 in OEIS). Its proximity to Euler's number e = 2.71828... follows from the fact that the integral can be approximated by the standard series for e. The difference is given by and also by ...

@r9m @robjohn ^^^

OK, one can say that a closed form can be got for that specific value of $a$, that is $a=\pi^2$ :-)

Thus

$$\int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x=F-e.$$ where F is Fransén–Robinson constant.

I later saw that one user specified it in the comments. I was aware of this constant since a long time ago.

Brennan.Tobias

hello

Chris's sis the artist

Overall, finding some nice infinite series for expressing such integrals is at least as interesting and nice as finding a closed form. Ramanujan did it again and again, read his notebooks.

r9m

@Chris'ssistheartist cool!! Seeing this for the first time here .. :)

robjohn

@Chris'ssistheartist Yeah, using that constant that is pretty much tied to the integral. Mathematical juggling

Brennan.Tobias

Do you know where a complete beginner could learn about these amazing integrals and sums?

r9m

8:01 PM

@Chris'ssistheartist it's there Ramy's notebook?

Chris's sis the artist

@r9m hehe. When I looked through the OP's questions I started from this idea that he met a similar integral to the posed one, and I was right ...

r9m

@Chris'ssistheartist that was sharp of you! :D (saved my rest of the night .. )

Chris's sis the artist

@Brennan.Tobias MSE is a good start. :-)

@r9m :D

@r9m Ramy's notebook? It's like the bible for someone that loves the area of integrals, series and limits. :-)

plouffe.fr/simon/math/Ramanujan's%20Notebooks%20I.pdf

r9m

@Chris'ssistheartist I mean the integral .. did you mean it's there somewhere in his notebook?

Chris's sis the artist

@r9m I don't remember I saw it there.

r9m

8:06 PM

@Chris'ssistheartist 'kay .. I'll dig through it after my exams then .. :-)

Chris's sis the artist

@r9m I only said that Ramanujan used to express ugly integrals using very nice series (it mattered less our present expectations about closed-form).

@r9m Wish you success then. :-)

r9m

@Chris'ssistheartist ah! I see .. :)

@Chris'ssistheartist I ain't gonna work with it anymore .. the appearance of a new constant out of the blue was enough for me :P

Chris's sis the artist

OK, put a name to the constant you get and you have a new closed form (in general, when one is tempted to say there is no closed form). Case closed.

r9m

lol .. didn't mean it like that :P (I had enough for a night .. that's what I meant)

Chris's sis the artist

@r9m :D

@r9m Do you want me to say a funny thing? :-) Well, related to the story with my series, I was thinking you got the closed form, but you wanna get a nice answer, and when I ask you how you did it you then you tell me you did it nicely without Mathematica stuff. :D

r9m

8:12 PM

@Chris'ssistheartist sure .. I like funny things :P (even at the cost of self-humiliation :P)

Agawa001

can someone make a notebook ? i mean a not-a-book :p ?

Chris's sis the artist

@r9m lol, not self-humiliation :D

@r9m Sorry for my English above, I might have put it better. :-)

@r9m that series has a special evil smile! Every time I think of it I'm about to laugh a bit. :-)

@r9m seriously speaking, people pay a lot of money to watch comedy movies, to get that entertaining stuff, but I don't remember a moment when I had more fun than in the period I did a lot of mathematics!

Never try to imagine me being that kind of serious face doing math, I'm very serious about doing math, but I have a lot of fun, mathematics makes me laugh a lot! :-)

@Brennan.Tobias Also try Paul Nahin's book, Inside Interesting Integrals, you'll have a good time with that book.

Brennan.Tobias

cool :)

Chris's sis the artist

@Brennan.Tobias It's not about rigourness, but about techniques in integrations, and the book is very friendly with new people in integration, not the could language of mathematics that might give you creeps.

Brennan.Tobias

sounds really fun I'll try it out

Chris's sis the artist

8:27 PM

@Brennan.Tobias You'll be very relaxed with that, no stress at all. I personally like very much the idea of writing such a book.

Never hide the entertaining part of mathematics.

Agawa001

i downloaded that notebook because of magic square

really nice

Chris's sis the artist

Yeah

@Brennan.Tobias Then, you can try Ovidiu Furdui's book. This one is also very nice but you need to know some stuff.

Brennan.Tobias

thank you

Chris's sis the artist

amazon.com/Limits-Fractional-Part-Integrals-Mathematical/dp/…

@Brennan.Tobias you can learn a lot, not only integrals, but also nice series and limits. It treats more the integrals with fractional part, but this is a class of integrals pretty cool.

r9m

@Chris'ssistheartist perfect math-head you are :P :D

Chris's sis the artist

8:36 PM

@r9m :D

@Brennan.Tobias I would recommend that you try to come up with your own solutions after a while, if possible, for each problem in these books. It's a good way to learn stuff.

Inside Interesting Integrals

Irresistible Integrals

@Brennan.Tobias also see the last one

Brennan.Tobias

oh fantastic :)

Ainz Ooal Goal

@Chris'ssistheartist And soon there will be your book there too :-)

Chris's sis the artist

@Brennan.Tobias Victor Moll also published another 2 books I don't have, they are more in the spirit of some work they previously published in some papers.

@AinzOoalGoal Hope so. :-)

user174558

8:52 PM

@Chris'ssistheartist As irresistible as Monica.

Agawa001

i just use amazon to pick on books i torrent-request em afterwards

Chris's sis the artist

@Brennan.Tobias I find those papers a very good stuff to study although at first sight one might say Ah, nothing very special. There is much power in those papers. They remind you of some very useful known formulae plus some modified forms for some integrals (again, known).

user174558

@Agawa001 Torrent? Have you heard of Genesis Library?

Chris's sis the artist

@JasperLoy :D

robjohn

@Chris'ssistheartist I edited those comments so that they don't take as much room.

user174558

8:54 PM

@Sajindia What do you wanna talk to me about? I hope you are feeling better.

Chris's sis the artist

@robjohn Thank you.

@robjohn I should have used [text](link) - if this still works.

Agawa001

most of links are dead there :(, nice place thu

user174558

@robjohn I am not complaining, but this is a peculiarity of this room. In other rooms, these things are encouraged to expand, hence the feature in the first place.

Agawa001

i would torrent chris's book too :p

Chris's sis the artist

@Agawa001 :D

user174558

8:55 PM

@Agawa001 Have you tried libgen.io

Chris's sis the artist

@Agawa001 You don't find me yet on any torrent. :-)))))

user174558

@Agawa001 I cannot find torrents for most books.

user174558

@Chris'ssistheartist Can we also torrent Monica?

Agawa001

wow last one was very helpful, to my fav immediatly :D

Chris's sis the artist

@JasperLoy hahahaha, that was good! :-)))))

user174558

8:57 PM

@Chris'ssistheartist I talked to myself for 30 minutes just now. I feel better now.

user174558

But I sent Jonas two emails and he has not replied. Sad panda...

Chris's sis the artist

@JasperLoy Nice. Maybe I should do the same, but the problem is that I prefer to think of some math more than spending time talking to myself (especially in this period of time). :D

user174558

@Chris'ssistheartist Actually, I talk to myself multiple times a day. It's my own psychotherapy.

Agawa001

@Chris'ssistheartist if u have some of these clients in ur laptop, close em immediatly, they ar a perfect egress for system-penetrators

Chris's sis the artist

@Agawa001 I don't have. Anyway, while on internet you're not safe as a general rule, if you have some important data to protect.

user174558

8:59 PM

@Chris'ssistheartist @robjohn You might as well edit this link as well then. =)

robjohn

@JasperLoy there are a lot of extended conversations with rendered mathjax. It is best for those to be able to look back as far as one can.

Chris's sis the artist

@JasperLoy I heard it's a powerful tool for healing.

user174558

@Chris'ssistheartist If I did not talk to myself, I would be dead by now.

robjohn

@JasperLoy If there are a number of expanded links like that together, it reduces the amount one can see back without having to scroll.

Chris's sis the artist

@JasperLoy I believe you.

robjohn

9:02 PM

@JasperLoy are you an only child? (no siblings)

user174558

The book I got in my mailbox today is 'A first course in mathematical logic and set theory' by O'Leary. I will start reading it on New Year's Day.

user174558

@robjohn Yes, but in this case, even if one has people to talk to, one should still talk to oneself.

user174558

It is a beautifully typeset book with two little birds on the cover.

robjohn

@JasperLoy Only children often talk to themselves.

user174558

@robjohn Or mentally ill people.

user174558

9:04 PM

By the way, for those interested in Zorich's Mathematical Analysis, a new edition will be out soon.

user174558

And for those interested in Gratzer's More Math into LaTeX, a new edition will be out soon too.

robjohn

@JasperLoy I bet the mentally ill people are not talking to themselves, but to perceived others.

user174558

@robjohn Aha, that is the schizophrenic people.

user174558

Munkres's Topology is very expensive. One should get Willard's General Topology instead for a thorough treatment of general topology.

user174558

I have no idea why Munkres is so popular, other than it being a classic.

r9m

9:09 PM

@robjohn plenty of examples among grown ups too .. some people can't leave that habit as they grow up .. (just like reading by pronouncing every word in mind .. )

user174558

@r9m Pronouncing

user174558

@r9m Also, one dot is for full stop and three for ellipsis, no such thing as two dots.

r9m

@JasperLoy thanks .. !

user174558

I think two dots arose online or in SMS.

r9m

@JasperLoy . .. ... .... ..... ...... ... :P

user174558

9:12 PM

Lang likes to reference Greenberg's algebraic topology book. I think because he is his student.

r9m

@JasperLoy who is Monica?

user174558

@r9m Monica Bellucci is an Italian actress.

Chris's sis the artist

@r9m A cute girl. :-)

r9m

@JasperLoy ah!! :D

user174558

@r9m She likes Monica, but I like Laura Ramsey.

Mitch

9:14 PM

@JasperLoy I found munkres inscrutable. needs more pictures.

user174558

@Mitch I think it already has many pictures.

Mitch

Oh.

user174558

@Mitch I prefer no pictures at all, because I draw them myself.

Mitch

Not enough.

for me

user174558

I think pictures should only be there if words are not clear.

r9m

9:16 PM

@Chris'ssistheartist Have you seen this recent article by Furdui? :-) We dealt with similar problems in past I think :-)

user174558

A word is worth a thousand pictures.

Mitch

Not to be antisolipsistic, but when you read topology, what is really going through your head is pictures.

user174558

Yes, but a word is precise while a picture is not.

user174558

Pedagogically pictures are important, but one should draw them oneself.

Mitch

but the picture communicates things more quickly

Chris's sis the artist

9:17 PM

@r9m No, I didn't.

user174558

@Mitch Yes, but it is against my taste.

Mitch

I don't think you taste very good either.

user174558

Yes, you are right.

Mitch

ha ha

user174558

Wait, what did you mean?

user174558

9:18 PM

Are you a cannibal?

Mitch

actually with algebra... yes, there arent exactly pictures going on but... but there is something visualized.

r9m

@Chris'ssistheartist 'kay :-) I kinda remember doing the first problem .. (in a discussion in chat)

Mitch

it's not just pushing symbols around.

Chris's sis the artist

@r9m Yes. Do you remember when? We did more series with $\psi$.

r9m

@Chris'ssistheartist I blogged about it back in 2014, so must be around then. Lemme check.

Mitch

9:20 PM

in English Language & Usage, Sep 15 at 14:09, by Mitch

Cannibalism jokes are funny because eating people is wrong

bbl

user174558

@Mitch I think in history there were people who ate dead people to survive.

r9m

@Chris'ssistheartist @Robjohn awesome solution

Chris's sis the artist

@r9m But aren't these problems different?

r9m

@Chris'ssistheartist yes .. but the basic principle was the same .. we were doing Able Summations :)

Vrouvrou

any idea for this please: math.stackexchange.com/questions/1546451/…

Chris's sis the artist

9:26 PM

@r9m Yes. Then there is another thing: he also used the Flajolet's paper. Initially I thought he avoided it. Did I tell you I calculated all series in Flajolet (I know the statement seems crazy, but I did it)?

@r9m I also have for my book such versions.

r9m

@Chris'ssistheartist yes you did, I remember! :D And I gasped with disbelief :P

Chris's sis the artist

@r9m I had no idea of this paper.

@r9m I know, it's hard to believe it. :-) Just wait for my book.

r9m

@Chris'ssistheartist neither did I until today noon :)

Chris's sis the artist

@r9m I know. It's so crazy to believe it. I don't condemn you at all, you're right to show disbelief, I might also do it if I were you.:-)

r9m

@Chris'ssistheartist I'll brush my teeth with your book :P (once I get ahold of it that is .. )

Chris's sis the artist

9:31 PM

@r9m lol, that's a statement!!! :-))))))

@r9m One thing: all I say about mathematics is saint. :-) If I say anything about math, it's exactly like that.

Let's wait then. :-)

r9m

@Chris'ssistheartist I am waiting patiently :)

Chris's sis the artist

@r9m how you got that paper, btw?

@r9m I got it.

r9m

@Chris'ssistheartist researchgate (I am stalking Furdui as well :P)

Chris's sis the artist

@r9m Great!

robjohn

@r9m When I said "only children", I didn't mean only children :-) I still talk to myself when thinking deeply.

Ainz Ooal Goal

9:44 PM

@r9m How did you join it ? Do you have an institution email ?

r9m

@robjohn :D

Chris's sis the artist

@r9m Just saying that sometimes life surprises you so nicely.

r9m

@AinzOoalGoal yas

Ainz Ooal Goal

lucky

r9m

@Chris'ssistheartist :-)

Chris's sis the artist

9:47 PM

Anyway. I learned many important lessons in the last period of time.

r9m

@Chris'ssistheartist who is not Romanian?

@Chris'ssistheartist who? who? who? :D

Chris's sis the artist

@r9m Not a known person probably for you. Anyway, it's premature to say anything about it.

robjohn

:25697492 do you have a publisher?

Chris's sis the artist

@robjohn Yes.

robjohn

@Chris'ssistheartist great! Have they given a deadline?

Chris's sis the artist

9:51 PM

@robjohn There is a deadline too.

The way it is.

Anyway, I learned many tough lessons (that almost drive you completely crazy), and I know exactly what to do in the future in all my mathematical activity.

My book is the proper response to everything, let its stuff quality define me.

r9m

@Chris'ssistheartist :D

:25697752 well .. pain builds character :-) (that's what I keep in mind when I face hard situations :-) )

Chris's sis the artist

Anyway. I don't say anything else.

@r9m Good.

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$Mathematics$ Mathematics