12:00 AM
Actually, @TedShifrin you have answered that for me.

@Alraxite But they are statements. Did I word my last message incorreclty?

@NikolajKyed Well, I didn't understand it.

Huh?

@NikolajKyed In any case, your question is based on an incorrect assumption if it doesn't specify some particular statement for D
And I don't think it does.
So, there.

@Alraxite I guess I should just try and work on it. One last question though. 6|(n^3-n). Am I supposed to read this as n^3-n is divisible by 6?

12:04 AM
@NikolajKyed Yes

@Alraxite Which is the sme as 6 divides n^3-n?

@NikolajKyed Yes. Both mean n^3-n is an integer multiple of six

@Alraxite Thanks for the help :)

@NikolajKyed glad to be of use.

@Ethan What are you doing now?

12:16 AM

@Ethan I am going to sleep, see you in my dreams.

night

12:54 AM
anybody know any good lagrange multiplier problems?

@alexander nope. let's make up a good one

@MikeMiller the following are the rules
logs, roots, polynomials, and exponentials only - no trig

is that the only rule

and must be answerable, not like a "there exists a solution" sort of thing

@alexander how about this. find the point on the $n$-sphere farthest from the origin in the taxicab metric.
or find what the largest distance is

12:59 AM
Hah

@MikeMiller hmm that's interesting

one can of course generalize this to looking at largest distance in the $p$-norm for various $p$, of $n$-spheres (w/r/t some other $p$-norm) to the origin

so we'd have $\mathcal{L}(\overline{x},\lambda)=\sum_ix_i-\lambda\left(\sqrt{\sum_i x_i^2}-1\right)$

if you don't wanna use fancy language you can do another such example by "what's the greatest and smallest distance from the origin of points on the unit cube? prove that there's a point of any distance between them." that's phrased about as poorly as is humanly possible but I bet you get my gist.

@Alex: Nice to see you again!

1:02 AM
@TedShifrin, how's it going?

@alex i don't know the answer off the top of my head and I can't focus cuz the guy behind me on the bus is playing his music real loud-like.

Just making enemies as usual ...:)

@MikeMiller yeah, i like that. let me make sure the solution works out nicely enough.
this would be used to torture calc students so i should make sure an answer exists first :-P
@TedShifrin ohhh yes, that's right

the second question has a geometrically obvious answer: 1 and $\sqrt{n}$ (distance to $(1,0,\dots,0)$ and $(1,1,\dots, 1)$ resp)
then one picks a path between them and appeals to the intermediate value theorem.

1:05 AM
Oh, I forget. But was trying to get this nee guy Rene Schipperus to learn to give hints rather than showing off with his PhD. The kid ultimately backed me up after. :)

@MikeMiller i wonder if there is a way to make the answer a little less obvious without making the lagrange method much harder

I'm losing my battles, @Kevin ...

@Ted You've been losing for a long time now. :-( Thankfully you're winning in the classroom as far as accepted pedagogical practice goes.
or well at least here you are. Can't speak to other places.

Only with students who really want to learn :D

@TedShifrin it really boggles my mind that that's such a rare thing in college

1:08 AM
@alex that ones only easy because we have a good geometric picture of the cube. pick anything else and it won't be so obvious even of the answer is identical.

Sad and depressing, @Alex. Largely society's fault.

@AlexanderGruber @Ted I need to run an ethnographic survey on my students to figure out why they all decide to come to an engineering school and study engineering
given how reticent they are of doing math

Why did I mention Alex earlier ...?

reticent isn't the right word.....reluctant perhaps?

@KevinDriscoll because they want a job

1:10 AM
Thinking about such a survey I realized that if I did it with IRB approval I'd probably lose my job...

Cuz they've been trained that the calculator will do it?

without*

I wonder if I can video my class in fall without approval? Last spring they all jumped at the idea ...

i think that the problem is we're teaching math like it's pure math but we're presenting it to them like a practical skill

We are?

1:12 AM
I heard the argument the other day that kids needn't learn multiplication tables anymore because a calculator can do that for them

@TedShifrin well, that's the justification for forcing them to take it, generally

while I understand and appreciate any push to make math teaching less rote, I would shoot myself if I needed a calculator whenever I wanted to find $4\cdot 6$

I think teaching proofs by rote is as bad as formulas by rote. Let's make them analyze and solve ... And what they hate most ... Write.

@MikeMiller solution: become an mathematician, refuse to do any math that isn't letters.

they only hate writing because they're taught to hate writing imo

1:16 AM
if i had the freedom, my ideal class for teaching non-majors would start with a couple months of python (or make it a prerequisite) and motivate most of what we're doing through mathematical modeling

@TedShifrin I have to disagree with you there, I love that my university makes me prove everything and practice "just doing" nothing. What I hate is no mark schemes.

I'm not talking about math major or hard classes. I'm talking about the generic low-level curriculum.

I want to know if I got it right, or if I accidentally missed something. As I've progressed I've gotten better, I cover proofs when I read books and have a go, but I like being able to see if I was right.
Oh

it's kind of hard to justify telling people that calc will be directly useful to their lives when all the examples we're giving them are like "Paul's business brings in $-t^2 + 32t$ dollars per month"

I make my students do plenty of proofs ... And applications, too.

1:18 AM
@AlexanderGruber i don't believe calc will be directly useful to the lives of the vast majority of students who take it

@MikeMiller maybe not, but i think it could be, if it was taught in the right way

I was going to give (cough) math.stackexchange.com/questions/827382/… this example. Here I wrongly applied liner algebra and messed up, but I wouldn't have spotted I messed up without seeing the rest of the question, I ask questions like this because answers often show me how to consider it so I wouldn't do silly things like this.

Agreed, @Alex. Bit real applications get unwieldy in a hurry, even with spreadsheets :D
The analysis/thinking skills is what they should take away, @Mike. Not the second derivative test.

@TedShifrin the gap between the math we do on paper and real applications isn't ever bridged either... I think that's why a lot of students don't want to learn it, because it seems like the practical application is hyperbole

@MikeMiller I have a few friends who are postdocs in Canada, and they say that students there are very accustomed to using calculators and having them available at all levels. Multiple times they were teaching calculus, and the fact that 4/24 = 1/6 (for example) were as detrimental to their understanding as not understanding derivatives

1:22 AM
@Alec: you can only compute dir der by $Df(a)v$ IF you know $f$ is dif'able.

it's much harder to make intuitive examples if people are functionally innumerate

@TedShifrin I'm looking into proving this now. I asked the question because the linear map property I use "seems" sound. As in I didn't go "Wait, WTF?" until AFTER I'd done it.
I now know that in the example I'm traversing the tangent plane, not the function itself WRT the dir derivative, but the fact I tried the to do this has spooked me.

That's all in my book you said had nothing in it :D

I know, it's in front of me, page 92, but I've covered the proof.

Yup @mixedmath

1:26 AM
I can prove it USING the linear map properties. (Diff => dir deriv) I'm trying to prove it's not an <=>

You compute dir deriv FROM THE DEFN and then observe the fn isn't even continuous!

Surely if I have all the directional derivs it's differentiable!? But no! Why is the question, so I'm exploring! Without proofs covered up, hence my being a total jessie.

Even if all the dir ders are $0$, the function might still be discont! Weird, but true.

No that "dir derivatives not being continuous" thing was useful They can still point, just not smoothly! I'll look into that!

@mixedmath the number of people i've run into at that level in my school is pretty astouding too. it's a good state school, but many of the non-STEM majors can barely do anything, yet they're all in precalc trying to find the domains of composite functions or evaluate $\tan\left(\arccos(x)+\pi\right)$

1:29 AM
You can have dir deriv 0 in every direction but still not have a tangent plane.

Someone I'm tutoring does very well understanding the ideas of calculus, but keeps making mistakes like assuming that division is linear

I really have no clue how to teach in that situation, because that stuff matters so little compared to the earlier stuff that they don't know. yet somehow they all ended up in that class.

Yup, @Mike ... Should we bypass alg skills to do ideas or make them quantitatively competent?
These are all good questions for the new site.

@TedShifrin You should post one!
Actually I think one like that was already posted

Not me. You're wrestling. I'm out the door!

1:32 AM
I'm sweating, actually.

Overheated?

@MikeMiller the best weapon i've got against that is comprehensively teaching them to check their answer by plugging in a couple numbers whenever theyr'e not sure about something

No AC in my room, @TedShifrin

i spent an entire period talking about that last semester. ended up being a worthwhile investment

1:34 AM
in theory, yes, @AlexanderGruber, but how do you reality check a derivative?
ah, nevermind

We had no AC in the bay area when I was growing up

by doing $f(x_0+.1)-f(x_0)$
@TedShifrin I don't wanna hear it

@MikeMiller i mean for the algebra mistakes, like $\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}$

but hey, that could work too, $f(x+1)-f(x)$ should be alright for a lot of situations

1:37 AM
@Mike: Even with my own students I often remind them specifically of stuff we did last week or last semester. Quick review ...

something i'd like to do is figure out a way for my students to learn by teaching each other but i can't figure out how to organize it

@Ted Theres a new site?

i don't like the idea of putting people in groups for in class projects, because it's a little kindergarten-y

@AlexanderGruber Ya figuring out how ot encourage peer-learning is a really active and difficult problem
@Mike thanks

1:42 AM
@MikeMiller Although it's been in read-only mode for me for over a full day now

@mixedmath not for me, I posted a comment a few hours ago

@MikeMiller oh, interesting

@KevinDriscoll i was thinking about making a board where students could make up problems to challenge each other
with some kind of incentive

@TedShifrin serious question (I was pacing and thinking) suddenly I had a flash of "A-level intuition" which is "thinking about it" rather than asking "What things have I proved can I make a digraph out of and traverse to get to the result"....
@TedShifrin why can't I just consider $(\cos(\theta),\sin(\theta))$
then look at $f(\text{that})$ and see what it does wrt theta?
Wait WTF am I babbling about.
(I'm trying to port change of basis to this)

You need $t$ in there, @Alec. But that won't prove diff'bility.

1:53 AM
@TedShifrin in this question I'll have to prove it is not differentiable but I still have to find the directional derivatives. math.stackexchange.com/questions/827382/…

$D_vf(a)$ linear in $v$ needn't mean $f$ diffable, as I said twice earlier. @alec
If you find dir der are nonlinear in$v$, then of course you're done.

@TedShifrin the question asks me to find the directional derivative THEN show it is not differentiable.
I'm on the first part.
I'm trying to define/use $D_vf(a)$ without ... well I'm not sure, this shouldn't have me stumped (it could be a 3am thing....)
They look so similar. I'm going to try writing on paper rather than thinking! This is embarrassing now.
@TedShifrin I was being daft, I swear to you I am better than this! I have used $\lim_{t\rightarrow 0}\frac{f(tv)}{t}=\sin(\theta)$
(because $f(0)=0$)

@Ted whence the notation $C^\omega$ for analytic functions?

I'll be honest, that answer seems "too nice".....

@MikeMiller Hello Mr. I-have-a-cool-ass-bed.

2:18 AM
Dunno @Mike ... One step past $\infty$?
@Pedro: @Mike is complaining that his ass is overheated, not cool at all!
@Alec: Was the numerator $x^2y$?

@TedShifrin HA! Why?

Summer in Santa Clara, @Pedro

Hehe... I will be going during summer, again.
I do like snow though.

You will mildew in NJ

@TedShifrin I've been there already. =)

2:24 AM
I thought you might have forgotten ...

@TedShifrin math.stackexchange.com/questions/827382/… function is there.

It was wrong btw, it should be $\sin(\theta)\tan(\theta)$
or $\frac{\cos(\theta)}{\tan(\theta)}$ or any other mix
Now it makes sense though, because we have angles where it undefined!

No, it's defined for every direction.

@TedShifrin you've seen a graph of the function right?
It doesn't look like it is

2:32 AM
What is the function on the coord axes?

So what is the dir der in those directions?

But anyway @TedShifrin $\frac{t^3\cos\sin^2}{t^2\cos^2+t^6\sin^6}\frac{1}{t}=\frac{\cos\sin^2}{cos^2}$
=$\sin\tan$
Even

When $\theta=\pi/2$ your alg fails ... As we've established.

So for $\theta=0$, we're fine, for $\theta=\frac{\pi}{2}$ aahhh! @TedShifrin
I don't see why though @TedShifrin because for $\theta=\frac{\pi}{2}+\epsilon$ $\sin>0$ and $\tan<0$ so should be negative. BUT $\theta=\frac{\pi}{2}-\epsilon$ and we should get a positive result, which we do.
Remember I said earlier "I get spooked when I miss a case" - this is what I meant.

2:40 AM
No one says dir der has to be continuous in $v$

@TedShifrin no they wouldn't be, that's why I was happy with $\sin\tan$

That's cont except at $\pi/2$ ...
But the function is $0$ at $\pi/2$, I mean.

Undefined at $\frac{\pi}{2}$ seems... well quite reasonable. TBH. It' s hard to imagine sort of this "defined + shape" (@TedShifrin if you define $\theta=\frac{\pi}{2}$ to give a dir diriv of $0$ then it's still discontinuous)

Your algebra that gave that formula is wrong at $\pi/2$.

By "defined + shape" I mean.... it's hard because this isn't very "analytic" but rather than being "we don't talk about (0,0)" we can, but only along the axes)
Yes, but why!

2:43 AM
Write it out carefully.

@TedShifrin write what out? $\tan=\frac{\sin}{\cos}$

No, the limit.
When $\theta=\pi/2$ explicitly.

@TedShifrin seriously, write what carefully? No matter how I look at it (using the sincos one) we get $\theta(1-\theta^2)^2 / \theta^2$
There is a $1/\theta$ term.
@TedShifrin $\frac{0}{0+t^4}$

Just put in $x=0$, $y=t$ ...
Right. Done. Your other algebra assumed ........

No I can see (as I state in the question) math.stackexchange.com/questions/827382/… that the partial derivs are 0 along the axis (as the function is constant)
So I can see where you're pulling this $\theta=\frac{\pi}{2}$ should give a dir deriv of 0, I can see where you've gotten that from.

2:51 AM
You tacitly assumed $\cos\theta\ne 0$ in your earlier alg.
Look at it after sleep.

What I don't know is why I would have missed.... @TedShifrin .... OHH! When I factor out the $t^2$ - that wouldn't be there if $\theta=\frac{\pi}{2}$
@TedShifrin is $\frac{1}{t^7}$ actually better?

Except it isn't :)
Night! :D

@TedShifrin $\frac{\cos}{t^4}$?
$=\frac{sin(\theta)}{t^4}$ for $\theta=0$
~$\frac{1}{t^3}$
@TedShifrin I'm waiting for a washing machine as it is, so ha!
@TedShifrin $\frac{0}{t^5}$?
Oh I had no idea we defined 0/x to be 0. Or that I was even doing these things!

7 hours later…
9:42 AM
It's official, I hate coding.

10:26 AM
@Studentmath What are you coding?

The stupidest things, as usual.

Like?

Trying to code a specific linked list, but of course from scratch, can't use anything useful Java has to offer, vague complexity requirements..

Oh. I thought you were coding math.
In that case, I can't help.

That wouldn't have been so terrible..

10:28 AM
@Studentmath Says who?

Would've also worked with everything -but- Java..
@Balarka It could've been extremely tough. Probably harder than this. But wouldn't be so boring

Yeah, that's kinda true.
Coding math is quite fun.
Why are you doing Java anyway, @Studentmath?

This feels like grinding water for the sadism of the head of the course..
Wanted to take few coding courses as I never coded before, earn few points on the way and get rid of the 'coding courses' requirements for the math major.. So I took data base (which is nice) and this terrible sadistic course
I should really read reviews of courses before just signing in..

Can't you just sign out?

The final test is within a month

10:31 AM
Oh noes. What're you going to do then?
Do they require Java for majors?
Not much zombies today, eh, @Studentmath?

@Balarka They require two computer science courses. I'm gonna study as much as I can (the tests are -terrible-), and hope to get a good grade. Nothing else to do..

I am learning sieve theory. It's weird.

I know nothing of number theory, certainly none of that. Is it interesting weird or just weird?

@Studentmath Interestingly weird.
Weird trick theory everywhere. Tricky analytic manipulations.

Care to explain a bit? Wikipedia article about it is interesting though dull

10:42 AM
Yeah, wiki doesn't give any explanation.
Well, the main development of sieve theory is done by Brun. It's about counting twin primes, you see.

Greetings

Are you familiar with Eratosthenes sieve, @Studentmath?
The idea is kinda like that. In general the doal is to estimate a siften set of integers. For example, let $a_n$ be some sequence and $P(z) = \prod_{k \leq z, k \in \mathcal{A}} k$.
The goal is to estimate $$\mathcal{S}(x, z) = \sum_{n \leq x \& (n, P(z)) = 1} a_n$$
For example, take $a_n$ to be defined as $1$ if $n = p+2$ and $0$ otherwise, $\mathcal{A}$ to be the set of primes, and $z = \sqrt{x}$
$$\mathcal{S}(x, x^{1/2}) = \sum_{p \leq x} a_p$$
$p$ running through the primes. This is essentially equivalent to asking how much twin prime there are upto $x$.
The so-called idea of "sieving" actually comes from combinatorial and probabilistic grounds. For example, one might ask you to estimate $$A_d(x) = \sum_{x \geq n = 0 \pmod{d}} a_n$$
One can guess that $A_d(x) = \rho(d) X + r_d(x)$ where the main term $X$ is something like $\sum_{n \leq x} a_n$ and $\rho(d)$ then is probabilistically the characteristic of the masses of $a_n$ that come from $d|n$
In general this idea is pretty useful and is the heart of sieve theory. The technicalities come from estimating very large main terms and one reduces the weight by splitting up the sum and using inclusion-exclusion. In particular these methods are pretty elementary and can be understood even without a priori knowledge in analytic/algebraic number theory.
@Studentmath

11:12 AM
Sounds extremely interesting. Any good reading sites/books?
@Balarka

11:27 AM
Sure. Try Halberstam's books.
Sieve theory is extremely simple (although technical) but it's currently running very fast in analytic number theory. Zhang's result on TPC is based on some technical applications of sieves.

@BalarkaSen $$\int_0^{\infty} \frac{x}{(e^x-1)(e^{x+\log\left(\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\right)}-1)} \ dx=\frac{\displaystyle \zeta(2)-\phi\left(\frac{3}{5}\zeta(2)-\log^2(\phi)\right)}{\phi-1}$$

@Chris'ssis You must have deleted what you told to r9m since I cannot find it.

@robjohn I just finished a mega solution ... :-) You have to see it :-)
@robjohn 8 parts. I have no idea if one can find a nicer way. (I'd be glad)

@Chris'ssis I have to finish some stuff for work, but I will look into it later this morning.

11:42 AM
@robjohn Sure, no hurry. Please take a look at the whole proof when you have time. I need your feedback. :-)

12:42 PM
@Chris'ssis No need to write in nested form.
$\phi$ is much nicer.

@BalarkaSen The nested form was written on purpose.

It can be derived elementarily, no? Using partial fractions?

@BalarkaSen Partial fractions? How?

Oh, I missed the $x$ in the numerator. Sorry.

OK :-)

12:49 PM
yet, I think I can work out a solution using complex analysis.
Darn the internet.
$$\int_{0}^\infty \frac{x}{(\exp(x)-1)(\alpha\cdot \exp(x)-1)} \mathrm{d}x = \int_{1}^\infty \frac{\log t}{t(t-1)(\alpha t - 1)} \mathrm{d}t$$
The integrand on the right has poles at $t = 0, 1$ and $t = 1/\alpha$
So now draw an appropriate contour and use fundamental theorem of residues, @Chris'ssis
It's quite elementary this way.

1:17 PM
Suppose, you are an editor of a magazine. Everyday you
get two letters from your correspondents. Each letter is as likely to be from
a male as from a female correspondent. The letters are delivered by a post-
man, who brings one letter at a time. Moreover, he has a ‘ladies first’ policy;
he delivers letter from a female first, if there is such a letter. Suppose you
have already received the first letter for today and it is from a female corre-
spondent. What is the probability that the second letter will also be from a
I think the answer is 1/2. As, we have only two outcomes in sample space: FM and FF

@Chris'ssis and I missed em all ;P :P

Where is my reasoning wrong?

@Sush Well u can ask an editor !! ... and besides I find that editor feminist :P .. I support equality .. I find such ideas are very unappealing :P
lol

Am I right? Is that 1/2 or 1/3?

Meh, @Chris'ssis, you can actually do that elementarily.
Set $\alpha = 1/\beta$ and $$\frac1{t(t-1)(\alpha t - 1)} = \frac{\beta}{\beta-1} \frac1{t-\beta} - \frac{\beta}{\beta-1} \frac1{t-1} - \frac1{t-\beta} + \frac1{t}$$
You're not in your full form today, @Chris'ssis, giving away integrals that I can do elementarily =p. You need a good sleep and healthy food.

1:39 PM
@BalarkaSen lol :-)
@BalarkaSen What's the next step?
@BalarkaSen I mean doing all without make using of some software to figure out the form of that integral's evaluation ...

@BalarkaSen How, suddenly, did you get so fond of integrals?

2:01 PM
@ParthKohli He simply couldn't resist the beauty of my integrals. :-)

@Chris'ssis Can he resist your beauty then?

@Chris'ssis log-frac integrals.
@Chris'ssis No software has been used. I only have PARI/GP and it doesn't give partial factors. It's for number theoretic purposes.
@ParthKohli I am not.

@BalarkaSen well, you need to know how to arrange things there ...

@Chris'ssis "arrange"?

@BalarkaSen how do you integrate (for example) $$\frac{\beta}{\beta-1} \frac{\log(t)}{t-\beta}$$?

2:12 PM
Ah. I would've used CA (as usual), but I know of an elementary method.
Gamma-zeta integrals.
$$\zeta(s)\Gamma(s) = \int_0^\infty \frac{x^{s-1}}{\exp(x) - 1} \mathrm{d}x$$

@Chris'ssis Sure it does. $s = 2$ and a logarithmic u-sub.
$$\int_0^\infty \frac{x}{\exp(x) - 1} \mathrm{d}x = \int_1^\infty \frac{\log(t)}{t-1}$$
There you go.

@BalarkaSen no, no no ...

@BalarkaSen You misses a $t$ in denominator in the right side.

2:18 PM
Hell.
I realize that.
Well, when there's a problem there always is a solution.

@BalarkaSen Then you have another integral. In our case it's about $$\frac{\log(t)}{t-\beta}$$

$$\frac{1}{t(t-1)(\alpha t - 1)} = \frac{\beta}{(t-1)(t-\beta)} - \frac{\beta}{t(t-\beta)}$$
Now you do that.

@BalarkaSen how would you integrate $$\frac{\beta \log(t)}{(t-1)(t-\beta)}$$? Also this one is ugly $$\frac{\beta \log(t)}{t(t-\beta)}$$

$t - 1 \mapsto t$

The user I hate is leaving stupid comments all over the place.

2:23 PM
Seesh, why does everything gets so complicated all the time.

The worst kind of comment he makes is "+1 This is the best answer" when the answer is wrong, lol.

@Chris'ssis No, that one is easy.

@BalarkaSen Is it?

Use the zeta-gamma formula above

@BalarkaSen Does it work? How?

2:25 PM
$$\frac{\log(t)}{t(t-\beta)} = \frac1{\beta}\frac{\log(t)}{t(t/\beta-1)} = \frac1{\beta^2}\frac{\log(t)}{t/\beta(t/\beta-1)}$$
Now sub $t/\beta \mapsto t$

@BalarkaSen Then you also change the integration limits, don't you?

@Chris'ssis Oh, indeed. And I can't derive an equivalent formula for that changed limits without a bit complex analysis (in which case you can just modify the contour a bit)
So, in it's whole, it seems better to try to evaluate $$\int_1^\infty \frac{\log(t)}{t-\beta}$$
real analytically.
And I have no idea how to do that.
OK, this real analysis is not my thing. Gotta go, loads of work to do.

@BalarkaSen this one blows up to $\infty$ though ...

2:42 PM
Is there a name for the category-theoretical generalisation of the quotient group/ring?