can someone recommend me a movie?

the departed

@JorgeFernández Yes, watch Good Will Hunting or A Beautiful Mind, lol.

I am trying to understand a concept. Does anyone know what the statement "the primes are diophantine" means?

what's the context

sup y'all

hey alex

long time no se

Yo

@MikeMiller yeah man it's been a while

how have you been doing?

doing alright. keeping busy for sure

you in grad school now?

i guess it's technically another month-ish (classes start in october), but yeah

cool, where you goin?

i'm at UCLA

nice, that sounds like a fun place.

make terry advise you

there's a summer thingie here to prepare us for one of the tests we're to take, so i'm here already

haha

i have a conjecture that one of the things UCLA uses to screen applicants is a mention of terry tao in their application

if they say "tao" but don't have a good reason to, like an explicit interest in harmonic analysis or analytic number theory or some such, down the garbage chute it goes

@MikeMiller what do you think they'd do if an applicant stated that he converted to taoism, just to be prepared?

well, if it was an otherwise good application, maybe they'd have a phone interview to see if he pronounced it tao-ism or dhao-ism

how about taoism, like mayoism

the worship of mayonnaise

they would probably be very confused about the presence of that fact on the application

this must be why they did not accept me.

i'll never understand these application processes.

i think a lot of it is personal connections

i.e. letter writers and who you've talked to at the uni

by that logic ucla should have been the last place i got into :)

but hey, i'm not complaining

then again, i didn't do too well on the GRE and had only near-perfect grades

so that could also be why i think that

i don't think it's worth thinking about now that it's past

it's a lottery is really all it is.

@MikeMiller well, it is for now

true

it'll all come back again for post-docs / professorships

or sooner for fellowships

i've been working on getting some grants lately so i'm doing some meditating on the application thing

ah, i see

how are things otherwise?

How's your back?

@MikeMiller they're alright. I'm looking forward to the new year, the new grad students seem cool. and i like the lecturer i'm working with better.

@skullpatrol about the same as it's been

:(

that's cool. i've had the same experience with the previous years' grad students here - seems like a good place to be

when you say lecturer you're working with, do you mean teaching-wise or research-wise?

teaching-wise. i'm on a TAship.

what are you lecturing?

calc 1, pretty standard

they don't give the interesting teaching jobs to grad students. we are slave labor.

ouch

i'm on a TAship too, we'll see how fun it is

I actually like teaching

(well, the administrative BS associated with it is not fun but the teaching part is good)

but, this is the year to put together my advising committee, on the research side

I'm actually thinking about moving in the applied direction, there are some really interesting problems in genomics that use a lot of algebra

I remember you've been thinking pretty seriously about doing applied for a while now, yeah?

Is there a difference between enlarging and stretching a curve?

I am trying to show that all parabolas are similar. That being said, two objects are similar if they are of the same shape but different size. Therefore, I take 2 arbitrary parabolas and I have shifted them both to the origin. The last step is to enlarge one to become the other, and I am done.

I guess in this case, enlarging and stretching are both equivalent.

But the same can not be said about ellipse/hyperbola.

I'm not sure what the difference is :) what, formally, is "enlarging" and "stretching"?

Hmm, just a quick thought. I believe enlarging would mean that all points on the curve move at an equal "rate". However, stretching would imply that some points move faster than others and what not.

How does that definition sound?

doesn't sound like a definition yet :)

From high school I recall enlargements being defined by scale factors.

Enlargements/reductions.

Oh gosh, I think this is starting to click.

"an enlargement of a circle $x^2+y^2=a$ is a circle with radius $ar$ for some $r$... a stretch is an ellipse with foci ... and ..."

one has to define these things to prove anything about them!

I'm heading to bed now though

good luck!

Thanks you have helped. Good night!

@MikeMiller yeah i suppose

I miss doing both pure and applied at once, like when I was doubling in math and physics. I like taking the real abstract stuff and finding ways to put it into applied context.

the type of math i dislike doing tends to correspond with the type of math one can't do that with (e.g. foundations)

it's just difficult to find the right sweet spot... too far on the pure side and i'm dealing with questions i'm not interested in, too far on the applied side and nothing cool happens.

Greetings

hi

@AlexanderGruber Hi. How are you doing? :-)

@Chris'ssis Not too bad. How are you?

@AlexanderGruber I've been working on some questions, but right now I'm looking at an older proof of mine.

an integral?

@AlexanderGruber There I used an idea that I can successfully apply to another question.

Forget the part with the DCT, it is simply there.

I like that way because proceeding as you see, I managed to bring the proof to the high school level.

@Chris'ssis Hmmmm, I see.

@AlexanderGruber Do you enjoy doing limits, series and integrals?

@Chris'ssis No. ;)

But I understand why you do.

@AlexanderGruber Oh, I see ... :-)

@AlexanderGruber Why? :D

@Chris'ssis well, i have a short term memory impairment that i've had since birth. it makes it very difficult for me to take in large expressions, i forget what i'm looking at by the time I get to the end.

it's probably the reason I went for algebra instead of analysis. No problems reading stuff like $Z(G)$.

@AlexanderGruber Sorry to hear that.

@Chris'ssis it's okay, i'm used to it. ;)

the ideas are definitely cool though

I like when the expressions have geometric or combinatorial interpretations

Yeap :-)

I like that too.

@AlexanderGruber: So why can't you do both; pure and applied?

Hello

what it means a "local invertible function" ?

I am not sure why the links above are not formatted in the way I expected them to.

@Chris'ssis How many hours per day do you work? And how many days per week?

@rehband Well, you know, life gets in the way, but I try to work as much as I can, it's the thing I love at most.

@rehband When I'm asked "How are you?", I usually answer "I'm working". :-) I dedicate myself in anything I do.

@Chris'ssis Haha nice, I try to do the same :)

@Chris'ssis How much coffee do you consume daily?

@rehband I don't smoke, don't drink coffee and don't take the known energy drinks. I prefer milk & cocoa & honey.

@Chris'ssis Amazing. By the way, do you have an alternate solution to problem 1.57?

(The problem u mentioned yesterday)

I don't like the one in the book very much because it makes use of an inequality which I've never seen before, and to which I can't find a proof

@rehband I think that's the natural way to take. By the way, I saw a question you posted that reminded me of a nice question I attended in the past.

@Chris'ssis Which one?

@rehband $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ It can be done with the high school knowledge.

@Chris'ssis Ah okay. What question did it remind you of?

@Chris'ssis Really?

@rehband Yeah.

@Chris'ssis Wow. How?

@rehband See my solution here i.stack.imgur.com/9ZB0r.jpg. We can use the same strategy.

@Chris'ssis Thx

@rehband Welcome. (ignore the part below with DCT, it's something else)

@Chris'ssis Okay :)

@rehband First check the question and the solution in the link above. Everything is done elementarily.

@Chris'ssis I'm reading it right now :)

OK

@Chris'ssis Holy cow, that's clever (I'm not through yet, but the $\pi^2/6$ squeezing is amazing)

@rehband :D

@Chris'ssis Amazing

Thank you! :-)

@Chris'ssis Why did you split the original sum the way you did?

@rehband I chose things such a way to be able to use the limit in the first part and then to get $0$ in the last sum. It's not hard at all, just a matter of practice.

@Chris'ssis Got it, thanks! I'll try to remember that trick

Lastly, the terms are 0+0+0+ ... infinitely many times

@Chris'ssis Right

@Chris'ssis I'm gonna tackle 1.15 from the book now.

@rehband Does it have a solution?

@Chris'ssis Yes

@rehband Can you post the question? Some might not have the book.

@Chris'ssis Sure, you mean in the chat?

@rehband Yeah.

Yes, 1 sec

Let $\alpha$ and $\beta$ be positive numbers. Calculate $$\lim_{n\to\infty} \prod_{k=1}^{n} \Big( 1 + \frac{k^{\alpha}}{n^{\beta}} \Big)$$

@rehband Thanks. It's a question by Ovidiu Furdui.

@Chris'ssis Np

It should be something to do with e :p

@N3buchadnezzar How do you know? :)

gut feeling

@N3buchadnezzar Haha, nice

@rehband This is nice to do it without pen and paper. The starting point is $\beta=\alpha+1$.

After taking log, we see immediately that all gets reduced to a Riemann sum that is straightforward.

The 2 other cases obviously depend on the value of $\beta$ compared to $\alpha+1$.

"Immediately, straightforward, obviously" - such intimidating words.

@Huy Ignore me for avoiding being intimidated.

@DanielFischer: Good morning, do you have a moment?

A moment.

Or two.

@DanielFischer: chat.stackexchange.com/transcript/message/17304587#17304587

@DanielFischer: Actually, $\Omega \subset ]0,L[$, not $]0,1[$.

@Huy Cauchy-Schwarz gives you $$\left\lvert \int_o^L \left\lvert \frac{\partial u}{\partial x_1}(s,x')\right\rvert\,ds\right\rvert^2 \leqslant \int_0^L 1^2\,ds\cdot \int_0^L \lvert \partial_1 u(s,x')\rvert^2\,ds,$$ and $\lvert\partial_1 u\rvert^2 \leqslant \lvert\nabla u\rvert^2$ is clear.

@DanielFischer: Oh, Cauchy-Schwarz. I was trying with Hölder and other inequalities and totally forgot about that one. :(

@rehband btw, did you try $1.28$, $c)$? I was just looking at it. Try that without pen and paper. :-)

@DanielFischer: Are you familiar with Schauder theory? I might have some questions about it tomorrow.

@Huy No, I'm not, unfortunately.

@DanielFischer: I am wondering, we defined $W_0^{1,p} = \| \cdot \|_{W^{1,p}} - \operatorname{clos}(C_c^\infty(\Omega))$. Why is the norm of $W_0^{1,p}$ not just the norm of $W^{1,p}$? We used the norm $\|u\|_{W_0^{1,p}} = \| \nabla u \|_{L^p}$ instead.

The norms are equivalent, since the boundary values are $0$, thus you can dominate $\int \lvert u\rvert^p\,dx$ by $\int \lvert\nabla u\rvert^p\,dx$. Hence it's a matter of personal preference which norm the author takes. I've seen both.

@DanielFischer: I know that they are equivalent. But in general, if I define some space $A = \| \cdot \|_B - \operatorname{clos}(C)$, wouldn't you assume that the norm used on $A$ is the same as on $B$ if nothing else was explicitly stated?

Yes, it's more natural to take the norm with respect to which one took the completion.

@DanielFischer: To me, choosing the different norm is important because then we have the existence of a weak solution of the Dirichlet problem by Riesz' representation theorem. Is it even possible to show existence with the other norm? Since you state that you've seen authors taking the more "natural" norm instead.

Sure it's possible. It may be more work, I don't know.

Alright, I'll see if I can find a book where the author uses the natural norm instead. Thank you for your two moments.

moment: point or brief portion of time :-)

Hey, everyone!

What do you think of my new sig? Bang

What's $I$, @MatsGranvik?

complex number

after all 2+3*I+10 =12+3*I

Yea, that's what I thought.

This might help, @MatsGranvik.

It might have to do with one part of the complex number dominating as either $x$ or $y$ get very large in $z=x+iy$. I've seen the term complex infinity floating around as well.

Yes but mathematica says it is infinity not complex infinity.

wolframalpha.com/input/?i=2+%2B+3*I+%2B+Infinity

that link went wrong

wolframalpha.com/input/?i=2+%2B+3+I+%2B+Infinity

2 + 3 I + Infinity

= Infinity

Oh, I wasn't saying that it's equal to complex infinity! I was just noting that it was fairly noteworthy given the nature of the discussion of complex numbers. If you change a complex number to it's modulus-argument form, you get $z=r(\cos \theta + i\sin \theta)$. This shows that a complex number is finite for all $x$ and $y$. Since $\theta = \arctan(y/x)$, as $x \to \infty$, $\theta \to 0$ so $z$ is largely dominated by the real part and is denoted by (for convenience), $\infty$.

That seems really wishy-washy. It's still a complex number. Ah infinity, what madness do you have planned for us next?

It's Saturday. Happy birthday, @JasperLoy!

"Complex Infinity" denotes the limit of arbitrary complex numbers whose magnitudes get arbitrarily large.

@Khallil, what you said about real parts is totally wrong.

Hello Professor @TedShifrin

Nice to see you again, is your class keeping you busy?

It was merely bait for you to appear and correct me, @TedShifrin!

@TedShifrin: I intend to turn some notes into a textbook at some point. Do you have any advice what I should pay attention to?

Does anyone here know how, if possible, to turn the fraction 1 / (5x^4+5x^3+5x^2+5x-1) into a power series so I can find the formula for the n'th coefficient?

@mathh: Why would it not be possible? Use Taylor's formula.

Turn it into a Taylor Series, @mathh?

Jinx!

=_=

@mathh: It seems the coefficients abide a rather nice rule too, if you compute them up to order 4.

(that is around 0)

In case you were wondering, @mathh: $$f(x) = \underbrace{ \sum_{r=0}^{\infty} \dfrac{f^{(r)}(a)}{r!} \ (x-a)^r }_{\text{Taylor Expansion}} \overset{a=0}= \underbrace{ \sum_{r=0}^{\infty} \dfrac{f^{(r)}(0)}{r!} \ x^r }_{\text{Maclaurin Expansion}}$$

$f^{(r)}(a)$ is the $r$th derivative of $f(x)$ evaluated at $x=a$.

^_^

^_^

How are you doing, @Huy?

Studying some functional analysis for an exam and not really making great progress.

What about you?

Thanks!

Hello, can someone help on my comment here math.stackexchange.com/questions/885711/… :)

@TrogdorTheBurninator I liked the previous username better.

@BalarkaSen :O I forgot to revert it

@BalarkaSen Here it's changed back

It's not.

Ah, now it is.

It takes some time to be fully updated

yeah, i know

@Chris'ssis Thanks :D

@TrogdorTheBurninator ;)

I'm back from holidays yay now I can work

poor kid forced to go on holidays

@TrogdorTheBurninator Did you have some fun?

@Chris'ssis Not really :c I don't like hot places and the beach xD

So I slept a lot

:-))))

@Chris'ssis Here's the problem I was talking about : Compute $$\sum_\rho \frac1{\rho}$$ where $\rho$ runs through the nontrivial zeros of $\zeta(s)$.

Fun thing is that you don't need to assume Riemann Hypothesis.

But it'll probably be too easy for you.

@BalarkaSen mathworld.wolfram.com/RiemannZetaFunctionZeros.html

I know the answer, I am just asking how would you do it.

Doing some set theory but it's not for an exam or anything, @Huy.

@BalarkaSen I just noted you changed the statement in the meantime.

Are you from Germany, @Huy?

@Chris'ssis ? I just made some clarificational changes that's all. chat.stackexchange.com/messages/17305806/history

@Khallil: I currently live in Switzerland. What do you mean by "are you from ...?"?

Oh, I meant 'do you live', @Huy.

@Khallil: I see, then I anticipated correctly. Where do you live?

I live in the UK, @Huy.

@Khallil: Where exactly? I've been to England and Scotland a few times.

Oh, cool! To be more specific, Scotland.

I live near Edinburgh, @Huy.

@Khallil: Nice. I've been to Glasgow, also wanted to check out Edinburgh but the weather wasn't very nice so I decided to stay in instead.

Yea, it's usually really bad where I am. Summer lasts for about 10 days a year, haha! Where exactly do you live, @Huy?

@Chris'ssis Actually you can do it by the Hadamard product produced in Mathworld, but it's just too complicated that way.

@Khallil: I live on the LHS of the lake in Zurich.

Ah, that's awesome! I'd love to be near a lake. I'd just go there to let my bad feelings be taken away by the passing water. You're lucky, @Huy!

@Khallil How's set theory going?

@BalarkaSen Those sums are known, I have somewhere some paper about them ...

Yesterday, I was doing a few questions on the complement of a set until I decided to watch the ending of Yu Yu Hakusho, @BalarkaSen.

@Chris'ssis Ah, I'd like to see them. If you can find them, please let me know.

=)

@Khallil: I just moved here in July. Before that, I didn't have a direct view to the lake and now I do and I can tell you it is very calming. I suddenly enjoy things like tidying or ironing just because I can look at the lake every now and then.

@Khallil Thanks.

@BalarkaSen OK

@Khallil bangs head Be darned that Yu Yu Hakusho of yours.

@Khallil: And of course, it is within five minutes of walking distance. But I'm usually too lazy and just open my window.

I'm green, @Huy.

Green with envy.

happy birthday @JasperLoy

^_^

@Khallil not sick?

You know you truly love Yu Yu Hakusho, @BalarkaSen.

@Khallil: If you want, I can show you a picture. It might make you green permanently though.

Show me, @Huy!

Yeah, congratulations @Jasper

Nope, not sick, @BalarkaSenpai.

@Khallil: Just the other day, I had to get up a bit earlier than usually, so I took the following picture around 6am: i.imgur.com/b0VBjAs.jpg?1

Could someone check my proof here though :) math.stackexchange.com/questions/885711/…

@Huy You see, that would be a reason for me to get up earlier every morning.

In fact, it would be a reason for me to get up in the morning.

@Khallil: I shut the blinds the first few nights and one night I was so tired I forgot about it and was woken up by the sun in the morning. Ever since I haven't shut the blinds anymore because waking up by the sun is much more comfortable than waking up by an alarm clock.

@BalarkaSen Some time ago, an American mathematician wanted to introduce me in the story of the famous Riemann Hypothesis, but I refused at that moment. He believed much in me, but in the end I'm just a human being, I have enough satisfaction creating and solving limits, series and integrals, I don't wanna work for many years on something that can be just an illusion.

@Khallil: I'm pretty sure I haven't slept past 8am since that day.

I feel more serene by just looking at the picture.

@Khallil: This is what it usually looks like an hour earlier: i.imgur.com/6ShCpdM.jpg?2

It was a bit cloudy, sorry about that.

@Chris'ssis well, as Holmes said, doing a whole lot of thing messes your brain up. But the thing is that you don't have to work on RH if you want to be introduced to it. Several students study the theory involved, but not all of them tries to prove RH.

@Chris'ssis How's the book going ?

@TrogdorTheBurninator Well, I'm still working on many problems. This will take some time ... :-)

I bet the air is really fresh there, @Huy.

@Chris'ssis fact is that one has to read a lot of stuffs to study the theory carefully. but doing that'd throw all the integral & series stuff out of your brain and that'd be not good for you, as you are interested in integral, series and limits.

@Khallil: Incredibly, yes. You should come to Zurich at some point in your life, if you enjoy things like these so much.

@BalarkaSen Do you really need to throw that stuff out of the brain ?:D

It's just like in GOT, many integrals are slaughtered by @Chris'ssis and the books takes very long to be published :D ... Game Of TripleIntegralsAndSums yay

@TrogdorTheBurninator lol :-)

I need to finish some proofs (bbl).

@Chris'ssis yes. you can't do two things at once.

either you have to chuck out the integral and series stuff or you'd have to kick out the number theory stuffs.

these two are the least connected branch among all of mathematics.

even differential geometry is connected to number theory

@Chris'ssis I shall do 1.28c next :)

@rehband Apply Cesaro-Stolz.

@Chris'ssis Will do

@BalarkaSen I see your point.

@Chris'ssis =)

concentrate on what you want to do

@BalarkaSen :D

@BalarkaSen Good point!

I disagree!

ok. why though?

There are interconnections if you look for them.

@skullpatrol what are you referring to?