I have considered going to culinary school in the future, but alas, $$

lol^

I won't charge you as much, @Clarinetist :P

Haha

oh... errr...

Im thinking there is a rule you can do

$\log_3(8)$ is the exponent $y$ so that $3^y = $ what?

3^y=8

$\int{\frac{r^{3}}{\sqrt{4+r^{2}}}}*dr$

OK ... So how do you write $7$ as $3^?$ ?

$log_3(Y)=7$

But this is probably not the best way to do the problem, if you need to use the calculator to get the answer, because the calculator doesn't have a $\log_3$ button.

@TedShifrin Any thermometers you recommend?

Huh? @Maximilian

[for meat, obviously]

Nah, @Clarinetist, just a quick-read digital.

Ah k

you would do log 7/log 3 right? change of base

I'm trying to get rid of square root in the denominator from the integral above. To do so, I'm multiplying by the conjugate?. So

hello, @Ted

Yes. But let's go back to your original problem, @Maximilian. Take $\log$ of both sides.

hello @Balarka ...

Purely out of curiosity, I remember around 7-10 years ago Chinese students were complaining about how we don't have $\log_{b}(a)$ keys and we always had to use either $\log$ (for base 10) or $\ln$ (for base $e$). Have they made such a thing in the U.S. yet?

$\int{ \frac {r^{3}} {\sqrt{4+r^{2}}}} * \frac{\sqrt{4 - r^{2}}}{\sqrt{4-r^{2}}}$

ohh

@Clarinetist Ti-84 supports it, but I don't believe there is a button for it.

$log_3(x-2)=7$ or would i also put 7 into log form

it has no exponent and would be base 7?

no, @Clarinet, everyone knows to do the usual conversion.

@Clarinetist No need to have such a thing, lol

$log_3(x-2)=log_7X$?

Noooo @Maximilian. Just use $\log$ or $\ln$, no base.

What is $\log(3^x)$?

I have considered buying a TI-Nspire CAS to keep up with current times (still doing tutoring and have met some people with them). Not that I'd want to, but I would like to get to know the calculator...

@TedShifrin pressing buttons on my TI-48 (or whatever it is)

I forget how to use natural log, i will need to take the khanacaemdy class on that in a minute (I did a bunch on just logs and properties)

Don't do that, @Maximilian. The properties are the same for any base.

i just know ln is base e but other than that i don't know anything

$\log(3^x) = x\log 3$

oh, you are just taking the logs of them, not changing them into a log?

well... ill try that

@Balarka: I was about the refer that torus problem to you, but you beat me.

What is $\sqrt{4+r^{2}}$ * $\sqrt{4-r^{2}}$ ?

As I recall, you and I discussed various ways of doing that months ago.

$x=\frac{log 7}{log 3}+2$

Is it just .. $4-r^{2}$.

yes, @Ted.

Yes @Maximilian

we did.

@Owatch $\sqrt{a}\sqrt{b} = \sqrt{ab}$, $a, b \geq 0$

@Owatch no

Ugh..

Your right.

Just forget about the square root and distribute the two expressions

@BalarkaSen: How do you calculate the homology $S^2 \vee S^1$? Usually you use Mayer-Vietoris. You might write something about how you can use the same relative argument as above to do this calcuation.

I think we came up with a way to do it using excision (not excision with long exact sequence, which is just M-V), but I don't recall.

I just want to get rid of the square root. I don't know if I'm approaching the integral right. I could do that though.

@MikeMiller Yeah, maybe I could.

@Owatch Not what I meant, and doing that won't get rid of the square root.

No, @Owatch. You seem bad about making substitutions in integrals. That should be your first instinct, rather than doing an hour of algebra.

Nothing big is necessary, I just think the OP might be a little confused without a word or two.

PS : just so you all know, if it were me, I would just use simplicial homology to calculate homology of torus :P

I know it won't get rid of the square. .

I understood that part.

way simpler when you have small \delta-complex structures

HMmm.. ok.. i guess that makes sesnse

x=3.77

@Owatch: Your instinct after all these problems should be to put $u=\sqrt{4+r^2}$ to make life easier and see what happens.

I did start it that way, but changed my mind when I thought I could factor 4 + r^2

Which you can't. But then I spent some time there trying to find other ways to simplify it .

You need to do enough problems that you know with 99% certainty what the right approach is.

Guess I haven't done enough problems.

In fairness, if I change the problem to $\displaystyle\int \frac{r^2}{\sqrt{4+r^2}}\,dr$ then one needs a different approach.

So you need to have an idea of which method to use where.

k, done.

maybe i should add an answer here converting my comments into a well-organized answer, but i am ashamed as it might look like hand-waving compared to John's explicit map :P

@Balarka: You are vain, but please correct the spelling to vein :P

whoops

you are slowly turning into Jasper, @Ted :P

I don't find that an insult, @Balarka.

I just hope my mental state doesn't resemble his :(

yeah, me neither

nice music

I hope he's OK.

Haha, hate to say... I am not a fan of Yiruma. Extremely overrated

me too, Balarka.

Ha, Eric asked me about that question last night, and Kevin and I suggested not to bother, because who the hell cares about joins?

I guess he wasn't satisfied with that. :P

When's the last time Jasper posted?

he is nice pianist though @Clarinetist

How is Eric doing, @MikeM? I haven't seen him on here in ages

yeah I didn't see him since long time

well, I gave him a lot of ingrediants for visualization in the comments

hehehe, I'm done with the proof to Knuth's problem!!!

@r9m ^^^ :D (less than 10 min)

so for $e^{\frac{3}{2}x}=5$, if i take natural log, then i can cancel the e and have $\frac{3}{2}x=log_e5$

from the impression I got from jasper he lives in conservative country I don't know.

I hope he is ok too

Yes, @Maximilian :)

@Ted: He's doing well, as far as I know! I beat him in poker last night.

Sweet

@KarimMansour Yes, but his compositions are just pop compositions, essentially. vi - IV - I - V is way used too often, especially in that piece you posted

He flopped the cards he needed, but I got mine on the river...

You brag whenever you beat people, Mike, but you are very quiet when they beat you :D

you guys play poker? how lame.

oh I see

I try not to have that happen very often, @Ted.

I don't, Balarka, but I'm going back to playing duplicate bridge when I move.

if you guys play chess then I would join

Well, I want to meet Eric finally, too. I've answered his questions on here way longer than I've known you or Pedro, @MikeM.

I don't like poker

I hate chess, Karim.

I play chess, @Karim

@tedshifrin then multiply by 2, divide by 3 and get $x=\frac{2ln5}{3}$ correct?

Yea, chess is great

grr, @Ted

cool we should play it sometime @BalarkaSen

@r9m I have to admit it was a nice game to play!!! :-)))

Except for the typo with $\ln$, yes, @Maximilian.

lol

@r9m I wonder if Knuth has a better proof than what I have.

what do you think of this one @Clarinetist youtube.com/…

Awesome

@KarimMansour Here's what I usually listen to

Can anyone explain the Hausdorff space in simple terms? I know only the definition that it's a topological space in which 2 distinct points have disjoint neighborhoods. Why do we need to define such spaces, how do we recognize them?

nice I like this too @Clarinetist

At least you aren't a Rachmaninoff fanatic, @Clarinetist. One of my brilliant math students this year is a violist/violinist and he was annoyed that I am not fond of the Russians (Rach and Tchaikovsky). My dad was a 20th century composer, btw, Clarinetist.

@KarimMansour Meh

@Paradox: Most spaces you run across are like that.

@TedShifrin Yes, I do recall seeing a Wiki page :)

@MikeMiller seeing his last question, he seems to like to think about everything rigorously (like thinking about very boring, albeit important, point-set topology behind the proof of CW-pairs having HEP), so I am hesitating on posting an answer

@TedShifrin Russians are great, but I've always liked French music more

i don't think he likes visualization :s

well, yes, and the Austrians and Germans, too, @Clarinetist :)

I mean, if you're going to post an answer to just about anything, it should actually be rigorous.

Every human being ever likes visualization.

False, @MikeM.

True, @Ted.

algebraists don't

Many algebraists really do not. Hence my antipathy toward algebraists :)

popcorn

LOL @Clarinetist

Not human.

@TedShifrin there's also the statement that any finite subset of a hausdorff space is closed. How?

@MikeMiller haha

Well, show the complement is open, @Paradox.

I fear I've overly influenced Balarka.

not just you. the whole gang of geometers out there.

$log\sqrt[3]{29}=\frac{1}{3}log29$ true?

@Maximilian correct

but I will break outta all this when I get to know enough geometry to do arithmetic geometry. and that might take years.

Awesome!

You'll get to read my colleague Lorenzini's book, @Balarka.

@Balarka: I think I've tried to stress this a hundred times, but the kind of geometry you're gearing up to study now and that kind of geometry are completely different.

@KarimMansour In that video, the cadenza starts at 5:53 and ends at 9:57. It is the one of the most disgusting piano passages I have ever seen.

Yes, @Balarka, I hate to agree with Mike, but he's totally correct.

@MikeMiller Well, I guess I got to know some algebraic topology to do arithmetic geometry, don't I?

Nevertheless, you should learn multivariable calculus, manifolds, etc., before you turn into a formal algebraic type :P

typo. dang typo.

Uh, I mean, I guess?

It's definitely worth seeing homology and cohomology in their place before doing sheaf cohomology. But the flavors are completely different.

Can anyone suggest a good real analysis book? One that is relatively simple and easy enough to understand if I want to study it on my own?

@Paradox101 Hands down, Tao

Eh, who cares. I am having fun doing the kind of geometry I am doing right now :)

and $log16-logm=\frac{log16}{logm}$

yeah I didn't like that part @Clarinetist

Ah, the wonderfully simple $\frac{r^{4}}{4\sqrt{4+r^{2}}} - \int{ \frac{-2r^{5}}{8(\sqrt{4+r^{2}})^{3}}}*dr$

Sheaf (Cech) cohomology in many ways is naturally motivated by analysis, @MikeM and @Balarka.

complex analysis, yeah. I have heard that.

Nooo, @Owatch, you're ignoring my advice ... again. You made it worse.

Who knows, maybe you will never switch to arithmetic geometry... @BalarkaSen

@Ted: I know this, but it's easier to appreciate the algebra going on after seeing it elsewhere first. If you recall, I went in the opposite order.

@Chris'ssis that's okay ... but in other areas he was a irrational person with all sorts of nonsense beliefs :P

What, by using u as you told me to?

this one is nice too @Clarinetist youtube.com/…

@KarimMansour It is disgusting-sounding yet beautiful harmonically, haha

I learned Cech cohomology in a complex manifolds course before I took algebraic topology, @Mike. I survived fine :P

I also like music that has violin too

The entire section says to use Integration by parts. So I did, using u as $\sqrt{4+r^{2}}$.

@Chris'ssis okay !! :D

Oh hell ... well, I meant substitution, not integration by parts.

@Ted: So did I, but it felt very unmotivated in the context I learned it in. That's all I'm saying. But I agree: see the "Uh, I guess" above.

If you're going to integrate by parts, you want things to get better, not worse.

@KarimMansour @TedShifrin IMO, the most beautiful music ever written by a human:

@Chris'ssis game to play? :O

A Ravel-nik :)

I'm more of a chamber music fiend, @Clarinetist ...

Eh, this is how I take a long time on questions.

@Clarinetist Do you mean terence tao's analysis?

Yes @Paradox101

@PaulPlummer Possibly. But I am gonna pick up Szamuely's book someday and read it.

Are there any solutions available of it?

I can spent over an hour doing useless nonsense. And I can barely tell what to do.

If you're going to use parts, @Owatch, you need to put something with the $\dfrac1{\sqrt{4+r^2}}$ so that you can easily antidifferentiate it.

Ah dang. I grew up teaching myself orchestration and have always been fascinated by orchestral writing @TedShifrin

So my substitution point still stands.

oh

@Paradox101 You will be hard-pressed to find complete solutions for an analysis book

Spend some time trying IBP? NO. Should have simplified first. Simplifying? NO. Should just go straight to IBP. Doing IBP? NO. Should just substitute.

(Rudin might be an exception, but I wouldn't recommend it for self-study)

My dad did both, @Clarinetist.

have you an idea please

This integral can be done with IBP by splitting the numerator into $r*r$

math.stackexchange.com/questions/1286816/… }

My initial motivation for learning altop was to know about this (galois theory) <--> (covering spaces) analogy. and now that I have read some altop, I don't think my curiosity has died out at all.

@TedShifrin Yeah, I just prefer orchestral writing. Ravel has a very nice string quartet too :)

Yes I've heard about Rudin but I suppose I should tackle it after building the basics from tao's analysis?

Yes, and even Mahler wrote a piano quartet movement :P

also, the Bloch-Kato conjecture is on my list of to-read topics. but I guess that'll take years.

@Paradox101 I don't know, to be honest with you (since I haven't really done a deep study into both texts), but I can tell you that Tao is by far more readable.

Ok thank you.

I'll give it a try

@Paradox101 He has Analysis I and Analysis II. I think they were just updated in 2015. Make sure you have the most current versions.

The thing is, I tried integrating $\frac{1}{\sqrt{4+r^{2}}}$, I can do a u sub, and get myself 2r on the denominator. Then I'm stuck.

@KarimMansour Another recommendation

If any metric space is a hausdorff space then does that mean that a seq in a metric space can converge to at most point?

yes

@KarimMansour Make sure you turn your speakers up a LOT for the beginning. It is very quiet.

What do you need with that to be able to do the integral, @Owatch? That was the whole point of my original advice. Steal one $r$ from the numerator!

but you can show this from the definition of convergence in a metric space from scratch very easily

yes, @Paradox, but you should be able to prove that directly anyhow.

lol

Oops, @Mike beat me ... as usual.

I should retire from here too.

nah

@TedShifrin NOOOOOOOOOO

@TedShifrin please have you an idea : math.stackexchange.com/questions/1286816/…

@TedS D:

I'm done. I'm going on a walk.

very nice yeah I like this kinda of music much more than the first you posted @Clarinetist its perfect state of mind while studying

I'll split it later.

Check on your battery, @Owatch :P

I'm very frustrated. No jokes.

@KarimMansour A good chunk of Vaughan Williams' music is built from English folk tunes.

But I will check battery

@r9m well, like him, I strongly believe that I was blessed by God with the wisdom I have in the area of integrals, series and limits. Just think that I have no background in mathematics, just self-educated. :-)

oh I see

....

@Vrouvrou: You should easily be able to do one direction by yourself, if not both.

@KarimMansour Another recommendation. I highly recommend listening to all 16.5 minutes sometime, but this is my favorite section of this beautiful piece.

so now i need to prove $log_bx^n=nlog_bx$

@r9m I'm working on the paper I send to AMM.

It was proven in a video i watched on Khan academy

@Chris'ssis which one?

Proven using what as known facts, @Maximilian?

@TedShifrin i started by f surjective and i have no idea how to itroduce an open set to say that it's image with f is open ?

@r9m jstor.org/discover/10.4169/…

@r9m Knuth's problem you showed me. The proof is pretty short.

I know it has to do like b^x=x, then try to do b^?=x^n or something, trying to remember without going back fully

@Clarinetist will check it after listening that piece you sent me it is beautiful

@ted anything i guess besides the rule itself?

How do you use surjective, @Vrouvrou?

Just says prove it

LOL, @Maximilian. So use facts about $\log$ or $\ln$.

@Chris'ssis oo!! okay!! I'll put my solution in my blog in a few days:D .. there are too many series and integral posts in my blog ,.. maybe I'll try and write sth about number theory in a post :-)

@TedShifrin i do not know !

I hate logs:P Teacher just put them in for really no reason.

@r9m OK :-)

OK, @Maximilian, say $\log_b x = y$. This means ?? And so what can you do next?

They're important in many settings, @Maximilian.

$b^y=x$

@Maximilian Apply that to the proof

Well I'm in a trig class. he just gave it to us at the end and said it will be on the final:P

So what is $x^n$, @Maximilian?

and i haven't taken Algebra 2 in a few years

Write down the definition of surjective, @Vrouvrou, and use finite dimensionality. You shouldn't be studying the math you're studying if you can't do this without my telling you.

x^n=b^(y*n)

@r9m I also have some subjects on number theorey (that can be helpful in the area of some series), but it's not the moment to attend them now.

@KarimMansour Going to bug you with two more:

Parentheses, @Maximilian?

@r9m in the past I also attended a lot of linear algebra, abstract algebra and geometry. I'm out of practice with them.

@KarimMansour

And what's $b^y$ in the context of this proof?

nice :D yeah I need to collect big play list for new music been listening to my playlist since ages

@Clarinetist

@KarimMansour This one's not as nice-sounding but I think it's harmonically very interesting.

$b^y=log_by$

@Clarinetist this is the playlist that I always listen to while studying youtube.com/…

oh, I wanted to reply to "who cares about smash product" message of @Mike but I forgot all about it. anyway, the reply was supposed to be "all of my intuition on hopf fibration depends on $S^1 * S^1$, so I do care :P"

@Chris'ssis I thought about sharing a few problems from matrix based inequalities and such ,, but that'll take a long time and effort from my part .. so I have postponed my plans till the day I feel like moving my lazy ass :P

@r9m some people here believe that all I did in terms of mathematics so far was integrals, series and limits, but I did a lot of stuff from various branches.

hi @BalarkaSen@Chris'ssis@TedShifrin

so probably you're right

hello @Gato

Salut @Gato

@r9m @Rememberme said about me I'm a liar that I never attended complex analysis and other stuff like that.

Anyway, many of them will be ignored for ever.

(they are already ignored permanently)

@TedShifrin How are you ?

@TedShifrin Ever heard of Henry Cowell?

yes, @Clarinet.

Pretty well, @Gato, thanks, and you?

@Chris'ssis if you came up with the proofs you do without knowledge of complex analysis ... I'd be shocked .

@TedShifrin Great :) I remember watching a performance of his Dynamic Motion . Very, very entertaining.

Better leave me out of this, @r9m.

Smash products are essential when doing later algebraic topology. It's how you do anything with spectra. But you don't need to know that until then, so when people ask me, I generally tell them to not bother with those either.

@TedShifrin fine, I have exams this week.

I survived my whole math life without spectra, @Mike. I saw them in one course in grad school.

@TedShifrin I just remembered Hippa's meme .. that's all (sorry about that)

So, @Ted?

have heard about spectra. probably I'll never do them.

@r9m Well, yeah, @Rememberme especially referring to contour integration, residue theorem I suppose.

LOL, oh, @r9m. One of my former students wants me to find him that meme.

haha

@TedShifrin Your dad studied with Milhaud?!?! OMG

@TedShifrin Isn't it there on Hippa's profile page?

I'll do K-theory at some point of time though.

Which universities have the best graduate programs in mathematics in the UK?

@r9m The thing that annoys most is that if I consider all these guys that talk about me at a blackboard and ask them to solve, say 10 problems, to solve in group I mean, I'm sure they solve no problem (in one year).

He was a student of William Schuman, @Clarinet. I know he was good friends with Milhaud. I didn't realize he'd studied with him.

@TedShifrin "After holding brief teaching positions at Columbia and at City College, CUNY, Shifrin received a Fulbright for study abroad, and in 1951–52 he was a student of Darius Milhaud in Paris—another whose influence is felt in Shifrin's music, particularly in his scherzando style, which has something of the fleeting “bounce” of French composers of the neoclassicist tradition."

@Chris'ssis but what will that prove? (except for the obvious cold truth that people don't change over an year :P)

Oh, yeah, @Clarinet, of course ... the Fulbright was just before I arrived :P

@r9m I mean I would never talk about someone's mathematics without being extremely good. I personally would never do that even if I were there.

I'm often pushed here in all kind of quarrels without my will.

@TedShifrin You know in complex analysis, we have a decomposition $F(z)=(z-z_0)^pF_1(z)$ where $z_0$ is an isolated zero and $F_1(z_0)\ne 0$, does the power $p$ is unique ?

@r9m I never talk about the others's mathematics just out of blue. I mean I try to be as nice as possible.

@Chris'ssis we all abide by the rules we make for ourselves .. no point in complaining that others don't abide by the same :|

Yes, @Gato: Write the Taylor expansion centered at $z_0$.

@r9m Sure, I accept almost anything, but not be harassed so often (just for fun).

@Chris'ssis one little word makes life easier ... 'ignore' ?

@r9m This is what I did already. :-)