And then you're in a whole new world eventually.

Learn French, German, Romanian, Russian, Chinese (Mandarin), or Japanese.

From what I've noticed, those are some really useful languages.

Oh, I almost forgot Italian.

So you are saying do the recommended stuff?

Yes.

But find a way to do the recommended stuff that provides for the future you want.

and also, I want to have some time to do mathematics too. And even now I am scavaging for more time

PhDs all have a second language

You'll be 18 by HS Graduation.

the thing I love doing obviously

I speak Portuguese and I speak Japanese (on paper and non-fluently)

Languages are great. They trick your mind into working a different way.

I don't think in math in Portuguese, or anything silly, but having an expanded language base allows me to develop language skills in mathematics.

but what if I dont get anywhere in math by that time? I assume I will be somewhere along the lines of like homotopy theory or infinity operads.

Where then language and programming feels like speaking another language.

@JulianRachman If you're there, you're already beyond many of the applicants.

and I am worried that if I take a test to get into a graduate program or something, I will fail because I self studied

I had friends that were accepted into MIT and then chose other colleges.

They had AP Calculus, at best.

and self study my be my downfall

@JulianRachman Are you trying to go straight from Highschool to College Graduate School?

Because I don't advise that.

nah.

I was going to say that I would take a test to get into a higher mathematics course and fail.

If in 4 years, you at least have Calculus under your belt, apply for Undergrad at MIT

I think grad will kill me :)

Oh, I see.

No, go about it this way: Don't self teach.

Understand that you know how to solve certain problems.

Don't forget the intuitions you have.

But forget everything that you "know".

Follow the path to higher maths normally, but at your accelerated pace.

For your high school, talk to your councilor about placing out of Algebra, Pre-calculus, Highschool Calculus, AP Calculus

If you can place out of those before you graduate highschool and still have a year or two to spare, then find your local college

And see if your school will let you take a college math course at that college.

In my Probability and Statistics Course, there was this High School student named Steven.

Swell guy.

It's not out of the question for a high school student to be taking college courses early.

But it's unreasonable to expect a student to pop fresh out of high school with a strong formal background in higher level mathematics when they were self-taught.

What does :"every different subsets [a] " means?

@Karlo, you're going to need to provide context.

ok

@JulianRachman Don't kill yourself. If you progress at the rate you're going, you'll have plenty of time to excel safely in a math program.

I see. Thank you so much. It is really wonderful to here these things from a grad student. LOL. Now this is the kind of answer I need for my proposed site.

I personally don't think it's worth a whole site.

Not an SE at least.

R be a subset of A x A and let it be relation of equivalence .a belongs to A and [a]={b |b belongs to A,aRb}.A family R consists of {[a_{i}} from all different sets [a_i] where I is a index set.Prove R can be parted

R is now two things. An equivalence relation and a family. Are they both the same R?

@Axoren I guess I will be going back to learning calculus.

no the first R is the normal R

from then it is the fancy one

(first R means relation)

@Axoren I have to gain a better intuition for the integral because I am not good at integration.

ok. Gnight. I ahve school tmrw.

@JulianRachman Look into Riemann sums.

@Karlo, please check the right of the chat for how to use LaTeX in chat.

ok will edit

@JulianRachman That's all an integral is. It's a Riemann sum with really small partitions.

Infinitely small, in fact.

OK. thank you again. Good night.

I hope that we will chat again.

I'll drop in every so often.

L be a subset of $A x A$ and let it be relation of equivalence .a belongs to A and $[a]={b |b belongs to A,aRb}$.A family R consists of ${[a_{i}}$ from all different sets$ [a_i]$ where I is a index set.Prove R can be separated in parts

@Karlo, Let $R$ be an equivalence relation on $A$. Given an element $a \in A$, $[a] = \{ b | b \in a, aRb \}$. A family $\mathcal{R}$ consists of $[a_i]$ where $i \in I$ and $I$ is an index set. Prove $\mathcal{R}$ can be separated into parts.

Is that what you meant?

So, you want to prove that you can partition $\mathcal{R}$? The way you can do this is to actually give an example of a partition of $\mathcal{R}$.

I'm out, good night all

Has the riemann hypothesis been generalized to quaternions and octonions?

^^^ @BalarkaSen @Hippalectryon

So, the differential operator d/dy cannot act on e^x

@user153330?

Good one :D

Hi.

@ABeautifulMind hohoho hi

Hi pal.

Hey @Ab, any progress(with mental illness recovery)?

@Committingtoachallenge Still trying to sort things out. Still feel really bad. You need 3 letters to ping.

@Abe Do you have many people to support you in person, who actually understand?

@Committingtoachallenge I talk to my mum every day. She is still working and supporting me financially. I hope she does not fall ill.

@ABeautifulMind That's good. Does she understand though[Empathise rather than sympathise]?

@infinitesimal Are you Iceboy[SP]?

What did the doctor say about the swelling @ABeautifulMind?

@Committingtoachallenge Yes. I have some friends I meet now and then too.

@infinitesimal Just gave her antibiotics. Porbably just an infection.

Yes @Committingtoachallenge

@infinitesimal This name feels very Behaviour->Fundamental-esque

He has me on ignore.

@infinitesimal Oh really? Why?

Dunno

@Committingtoachallenge hi! how are you =) ?

@user153330 Feeling quite good right now :). How about you?

@user153330 Have you had previous names on here?

@Committingtoachallenge fine =)

@Committingtoachallenge nope

@Sawarnik Do you still come here?

@Committingtoachallenge please don't forget " I will likely do a write up on this and test it myself in the near future. I will have to go back and see how I do on old exercises when I am a few textbooks in. " and if you have more wisdom on taking efficient notes, it takes me 2 pages of text to make 1 page of notes which is too inefficient

I saw him earlier.

@user153330 I won't. It is definitely a relevant concern for doing self-study - and my challenge as well

@Committingtoachallenge ok thanks =) do you plan to enter after that graduate school?

@Committingtoachallenge yup :D

Greetings

@user153330 Do you have "Introduction to Finite and Infinite Series and Related Topics" by J. H. Heinbockel?

Greetings my friend.

@infinitesimal Hi

How are you @Chris'ssis?

@infinitesimal Looking for some books to buy now. Not bad, thanks. :-) You?

@Chris'ssis fine thank you.

:-)

@infinitesimal Interested in this one

amazon.co.uk/Introduction-Finite-Infinite-Series-Related/dp/…

I m back

Hi @Chris'ssis

Welcome

@SayanChattopadhyay Hi

Hey what kinds of books are these

@SayanChattopadhyay Those books I usually enjoy since they are about integrals, series and limits.

@Chris'ssis greeeting! =)

@Chris'ssis no, i'm also searching for it :(

@user153330 Hi. I'm going to buy it from Amazon.

:19846987 Cute :-) That one disappears too! :-)))))

Can anyone help me load mathjax onto my Tumblr site? I've been trying for a few minutes now with no success

@EricAuld have you seen webapps.stackexchange.com/questions/11904/… ?

Hi people! I have some difficulties in evaluating an integral of an affine function, over the intersection of the $n$-ball and a hyperplane. Could you take a look? Dear @DanielFischer, you solved a previous problem of that kind for me before, but I cannot follow your arguments in this case. Could you help on this? Apologies for interrupting you again...

@nullgeppetto That's right, you integrate over $\Omega\cap U$, since that's the only part of $\Omega$ where $f$ doesn't vanish. By translation, we can assume that $\mathbf{x}_0 = 0$ (the $b$ may change, $\mathbf{a}\cdot\mathbf{x} + b = \mathbf{a}\cdot(\mathbf{x}-\mathbf{x}_0) + (\mathbf{a}\cdot \mathbf{x}_0 + b)$), and by rotation, we can assume that $\mathbf{a} = a e_n$ is a multiple of one of the standard basis vectors. Thus you need to compute the volume of the cap $\Omega\cap U$,

and the integral $$\int\limits_{\substack{\lVert \mathbf{x}\rVert < r\\ x_n > b}} x_n\,d\mathbf{x}.$$

By scaling, we can of course assume $r = 1$, and we need only consider $0 \leqslant b < 1$, the case $-1 < b < 0$ follows by symmetry, for $b \geqslant 1$, the intersection is empty, and for $b \leqslant -1$, we have the full ball, which is a well-known case.

Now, let's denote the volume of the $k$-dimensional unit ball by $\tau_k$, then by Fubini, the integrals we need become $$\tau_{n-1} \int_b^1 \bigl(\sqrt{1-x_n^2}\bigr)^{n-1}\,dx_n\quad \text{and}\quad \tau_{n-1}\int_b^1 x_n\bigl(\sqrt{1-x_n^2}\bigr)^{n-1}\,dx_n.$$

Making the substitution $x_n = \sin \varphi$ may be useful for the first, I don't know off-hand. For the second one, we can write down the primitive $-\frac{1}{n+1}\bigl(\sqrt{1-x_n^2}\bigr)^{n+1}$ directly.

@DanielFischer, well, you know too many things off-hand! Thank you very very much honestly! So, I need to decrypt your suggestions first! I will try to do so, and I'll let you know. Do you want to add your solution under the question, or would you like me doing that, of course by acknowledging you?! Thanks again!

Any opinions on Functional analysis - Berezansky

Is there a meaningful notion of a complex random variable?

Or even one of complex entropy?

I ran into this problem when I was working on a feature selection method and noticed that one of my features was complex.

@nullgeppetto I'm lazy, too much typing to add an answer ;)

@DanielFischer, when I decrypt your words, I will do so! I also have to decode @TedShifrin 's answer first! Thanks both of you!

How goes the book @Chris'ssis

@Committingtoachallenge All is fine thanks. I'm working on some very nice problems and also get very nice proofs.

All will be marvellous at the end. :-)

@Chris'ssis I think I already asked you this, but is there an end goal? How will you know your book is done?

@Committingtoachallenge I plan to add about 300 problems, and I have major part of them in mind. I also have the proofs to them. Then I'll also probably add some questions from new research.

@Chris'ssis Oh very cool. Is this 300 problems with these proofs attached, or maybe most of these problems in one book unsolved and a solutions manual?

@Committingtoachallenge No, I'll add both the problems and the solutions to the same book. Then I'll try to turn all into a kind of story if I'll be able to do it and show the way things from all book connect to each other.

@Chris'ssis Will you leave any problems to the reader?

@Chris'ssis Oh that would be very cool, I actually have no idea how you would do that. I look forward to it

@Committingtoachallenge I might have some open problems, not sure yet about that. My aim is to offer to the reader a maximum amount of pleasure, and if possibly not to let there just the cold language of mathematics.

@Chris'ssis Sounds nice. I shall get back to work, good to hear from you and about your book :)

@Committingtoachallenge Thanks ;)

The second answer has been removed, it seems. It was at -2 when it was removed.

@Axoren Yes

@Axoren I removed my answer

@Axoren first while I was reading the answer, I saw two immediate upvotes. Then I upvoted, which made it 3+ .....after I answered, both of us got immediate 2 downvotes

@TheArtist That was definitely sketchy.

Then I deleted my answer (which works to the same logic as yours) .....and then I downvoted the question , which made it go to 2 upvotes

I don't know where the rest of the upvotes and downvotes came

@Axoren two immediate upvotes to the answer !!!!!

Yeah, but that one is a very innovative one, and there's a lot of people watching the question now.

The difference was that when the question was fresh, it got an abnormal amount of attention.

@Axoren yes but don't you see, the question got immediate 2 votes up , two answers got immediate 2 downvotes each, a new answer gets two immediate upvotes

Yeah, maybe. There's no way to tell even if we had diamonds.

The latter could just be coincidence.

But the former seemed suspicious.

@Axoren I don't think the answer right now works either

@TheArtist Why?

@Axoren The third point on the explanation: the Middle sister is someone who tells the truth at times and the other time shes a liar....there is nothing as answering randomly, she will tell yes (which makes her a truth teller) , or no (a liar).......

I doesn't say she is a truther or a liar.

It says she tells truths or lies.

She's not selecting a nature.

She's just selecting an answer.

Sometimes it's the truth, sometimes it's a lie.

@Axoren ya but what she tells can be a truth or a lie

Yes that's what I'm saying

Exactly.

If she answers at all, it works.

The Eldest can't answer because she might be lying.

And the Youngest can't answer because she might be telling the truth.

If you ask the liar (you don't know its the liar) she will lie .....if you ask the middle sister she lies (let's say) then ....what's the difference between the two?

The liar won't know if it's a lie until after I marry one of them.

So what if she's telling the truth?

@Axoren you only get to ask one randomly chosen girl

Look at my response to Joffen

@Axoren I'm not talking about your answer

It doesn't matter which girl I ask :D

Which answer, then?

@Axoren I know, our answers work....I'm talking about the answr which got +4 votes

@Axoren and all this time you thought I was referring to your answer -.-

That one works because of the nature of the question.

Question.

@TheArtist If you have asked the middle it matter not what she has answered, you will get one you want.

It forces the answer to be the correct one based on the askee.

@Committingtoachallenge is it okay if I answer you here instead of on the answer?

I have some confusing notation in this question.. $A,R$ are rings. $P,+$ is a group, $A,+$ is a group. Then they are talking about $S=P+R$. What do you think they mean? They treat it as a ring, but.. do they mean $S=P\cup R$?

@Axoren Your answer is assuming the middle girl chooses a yes/no randomly, rather than telling the truth or telling a lie randomly

@Committingtoachallenge She is the only one capable of answering regardless.

@Axoren She also can't answer. Is she was saying yes/no randomly she could

@Committingtoachallenge look , there are three girls...and you go and ask one girl , is the other girl elder to the other one...and she answered yes, who do you choose?

@Axoren She can't tell the truth or lie randomly without knowing which is the case

@Committingtoachallenge I didn't see that restriction in the text.

@Axoren It says she lies or tells the truth randomly

No it doesn't.

It says she tells the truth sometimes and lies the rest of the time.

@Committingtoachallenge There is A,B,C girls.... You go to B and ask is A elder to C? b tells yes.....who do you pick????

@robjohn I finally found a nice proof to the limit I posted yesterday (the one with double integrals).

@Committingtoachallenge It's only describing the results of her actions.

@Axoren How can she tell the truth or lie without having a truth or lie existent?

@Committingtoachallenge You're holding a banana weighing 10 kg.

@TheArtist You choose C

Did I tell a lie or a truth?

@Axoren The false value is existent

@Committingtoachallenge why C?

@Committingtoachallenge So, wasn't I lying just now?

I'm blatantly lying about you holding a banana.

@Axoren Yes, but that is an existing property. "Will I marry the middle sister?" Is about the unknown future

And I'm further lying about the fact that it's really fucking heavy.

@Committingtoachallenge You will be holding a banana in 2 minutes.

I guarantee it.

@Committingtoachallenge if you picked C your wrong, because C is the middle Girl...and B is the truth teller , and A is the liar

@Axoren Is that true or not?

@Axoren Tell me

@Axoren I will make the opposite reality arbitrarily :)

@Committingtoachallenge It's true, I guaranteed it.

However, if I were honest, I couldn't answer.

Because I wouldn't know if you would avoid bananas for 2 minutes to make me a liar.

@Axoren Say the middle girl feels like lying, can she?

Yes.

But now consider this, the "intentional liar" can't answer either.

@Axoren She can't lie, give that banana example again

@Axoren You guarentee I am holding one as a lie, and I can just hold one, and you were telling the truth

@robjohn without a careful approach that limit becomes pretty awful. I'll upload the file soon.

@Committingtoachallenge She can lie, but she can't choose to lie.

But that's outside the scope of the riddle.

It was never stated that she chose to lie, just that she did it some of the time.

@Axoren I will think about this again tomorrow, it's 1:11AM so I feel I probably can't appreciate your idea properly sorry

You need to a banana.

Hopefully you don't find one while sleeping, otherwise I was telling the truth.

@TheArtist Give me a sec to write this down

@Axoren I have bananas on my desk :P

@Committingtoachallenge Then I'm the middle sister.

Oh, well that's going to be awkward out of context.

No one star that

@Committingtoachallenge Ohhh I got it lol

@Committingtoachallenge no need to write it down...

@Committingtoachallenge got it ;)

@TheArtist Good because I am so tired I can't think

@Committingtoachallenge what a tubelight I am

@TheArtist Well I shall sleep now I guess :S

@Committingtoachallenge It's been two minutes, are you holding a banana?

@Committingtoachallenge goodnight

@Axoren As soon as you asked that I picked it up :P

Does that count

NOOOOOO!

xD

If anyone has any ideas I would be interested! stats.stackexchange.com/questions/126419/…

night

@robjohn I uploaded the files with the proof. I couldn't find anything better than that, but I think that way is pretty nice. And, yes, I worked for some time on that proof ... (initially I wanted to use all kind of ugly manipulations using triple integrals and then the swapping of the limit and the integral based on the uniform convergence, but I dropped that way after a while - things were pretty unpleasant). Just take a look at that when you have some time.

Back in 30-60 min.

@Committingtoachallenge Good night.

@Chris'ssis Are you back?

Hi @TedShifrin.

hi, Jasper. Just stopping by to see what nullgepetto and Daniel said. But I'm off to teach. Hope you keep the room under control :)

Hope I keep myself under control.

Well, it gives you direction to keep others under control :)

Hi@Chris'ssis

@ABeautifulMind @SayanChattopadhyay Back. Hi. How are you doing?

rejoice everyone, for today is a great day!!!!!!!!!!!!!!

I m doing fine@DonLarynx

How r u doing@DonLarynx

A great day why@DonLarynx

hi

@SayanChattopadhyay did you change your pseudo ?

The page with stats says that this site has average 577 questions per day. That's about 24 question per hour.

I can't help the feeling that at the moment the pace is much faster.

Probably we are know in the part of the day when the most posts are made.

Morning.

Evening @Mike

How's things?

Bad.

I hope one day I can say good in this room.

That day may or may not come.

That's a tautology, but I am sure it will Jasper

I'm okay I guess

@MartinSleziak Also, the stats are probably an average over $n$ days, and we had a big drop around the change of the year, as usual.

@DanielFischer That page says that it is average over past two weeks.

So the beginning of the year should not matter that much.