Mathematics - 2014-06-09 (page 1 of 3)

Alec Teal

12:00 AM

Actually, @TedShifrin you have answered that for me.

Nikolaj Kyed

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

Alraxite

@NikolajKyed Well, I didn't understand it.

Ted Shifrin

Huh?

Alraxite

@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.

Nikolaj Kyed

@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?

Alraxite

12:04 AM

@NikolajKyed Yes

Nikolaj Kyed

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

Alraxite

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

Nikolaj Kyed

@Alraxite Thanks for the help :)

Alraxite

@NikolajKyed glad to be of use.

Jasper Loy

@Ethan What are you doing now?

Ethan

12:16 AM

just reading some stuff

Jasper Loy

math.stackexchange.com/users/147263/words-that-end-in-gry cast 1885 downvotes and 0 upvotes, lol.

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

Ethan

night

Alexander Gruber

12:54 AM

anybody know any good lagrange multiplier problems?

Mike Miller

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

Alexander Gruber

@MikeMiller the following are the rules

logs, roots, polynomials, and exponentials only - no trig

Mike Miller

is that the only rule

Alexander Gruber

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

Mike Miller

@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

Alec Teal

12:59 AM

Hah

Alexander Gruber

@MikeMiller hmm that's interesting

Mike Miller

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

Alexander Gruber

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

Mike Miller

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.

Ted Shifrin

@Alex: Nice to see you again!

Alexander Gruber

1:02 AM

@TedShifrin, how's it going?

Mike Miller

@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.

Ted Shifrin

Just making enemies as usual ...:)

Alexander Gruber

@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

i forgot about your ping. what happened again?

Mike Miller

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.

Kevin Driscoll

@Ted The Crusade of "Hints only, please!"

Ted Shifrin

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. :)

Alexander Gruber

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

Ted Shifrin

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

Kevin Driscoll

@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.

Ted Shifrin

Only with students who really want to learn :D

Alexander Gruber

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

Mike Miller

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.

Ted Shifrin

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

Kevin Driscoll

@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

Ted Shifrin

Why did I mention Alex earlier ...?

Kevin Driscoll

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

Alexander Gruber

@KevinDriscoll because they want a job

Kevin Driscoll

1:10 AM

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

Ted Shifrin

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

Kevin Driscoll

without*

Ted Shifrin

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

Alexander Gruber

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

Ted Shifrin

We are?

Mike Miller

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

Alexander Gruber

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

Mike Miller

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 $4\cdot 6$

Ted Shifrin

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.

Alexander Gruber

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

Mike Miller

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

Alexander Gruber

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

Alec Teal

@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.

Ted Shifrin

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

Alec Teal

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

Alexander Gruber

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"

Ted Shifrin

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

Mike Miller

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

Alexander Gruber

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

Alec Teal

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.

Ted Shifrin

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.

Alexander Gruber

@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

mixedmath

@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

Ted Shifrin

1:22 AM

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

mixedmath

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

Alec Teal

@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.

Ted Shifrin

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

Alec Teal

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

Ted Shifrin

Yup @mixedmath

Alec Teal

1:26 AM

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

Ted Shifrin

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

Alec Teal

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.

Ted Shifrin

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

Alec Teal

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

Alexander Gruber

@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 $\tan\left(\arccos(x)+\pi\right)$

Ted Shifrin

1:29 AM

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

Mike Miller

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

Alexander Gruber

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.

Ted Shifrin

Yup, @Mike ... Should we bypass alg skills to do ideas or make them quantitatively competent?

These are all good questions for the new site.

Mike Miller

@TedShifrin You should post one!

Actually I think one like that was already posted

Ted Shifrin

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

Mike Miller

1:32 AM

I'm sweating, actually.

Ted Shifrin

Overheated?

Alexander Gruber

@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

Mike Miller

No AC in my room, @TedShifrin

Alexander Gruber

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

Ted Shifrin

Btw @Alec ... This chat.stackexchange.com/transcript/message/15981484#15981484 was for you !

Mike Miller

1:34 AM

in theory, yes, @AlexanderGruber, but how do you reality check a derivative?

ah, nevermind

Ted Shifrin

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

Mike Miller

by doing $f(x_0+.1)-f(x_0)$

@TedShifrin I don't wanna hear it

Alexander Gruber

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

Mike Miller

@TedShifrin How much time should we spend on material that should be already known by students?

Alexander Gruber

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

Ted Shifrin

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 ...

Alexander Gruber

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

Kevin Driscoll

@Ted Theres a new site?

Mike Miller

@KevinDriscoll just linked to it

Alexander Gruber

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

Kevin Driscoll

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

@Mike thanks

mixedmath

1:42 AM

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

Mike Miller

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

mixedmath

@MikeMiller oh, interesting

Alexander Gruber

@KevinDriscoll i was thinking about making a board where students could make up problems to challenge each other

with some kind of incentive

Alec Teal

@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)

Ted Shifrin

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

Alec Teal

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/…

Ted Shifrin

$D_vf(a)$ 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.

Alec Teal

@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 $f(0)=0$)

Mike Miller

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

Alec Teal

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

Pedro Tamaroff

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

Ted Shifrin

2:18 AM

Dunno @Mike ... One step past $\infty$?

What answer @Alec?

@Pedro: @Mike is complaining that his ass is overheated, not cool at all!

@Alec: Was the numerator $x^2y$?

Pedro Tamaroff

@TedShifrin HA! Why?

Ted Shifrin

Summer in Santa Clara, @Pedro

Pedro Tamaroff

Hehe... I will be going during summer, again.

I do like snow though.

Ted Shifrin

You will mildew in NJ

Pedro Tamaroff

@TedShifrin I've been there already. =)

Ted Shifrin

2:24 AM

I thought you might have forgotten ...

Alec Teal

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

Ted Shifrin

That's why I asked. Your answer be wrong. Check your algebra.

Alec Teal

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!

Ted Shifrin

No, it's defined for every direction.

Alec Teal

@TedShifrin you've seen a graph of the function right?

It doesn't look like it is

Ted Shifrin

2:32 AM

What is the function on the coord axes?

Alec Teal

0

Ted Shifrin

So what is the dir der in those directions?

Alec Teal

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

Ted Shifrin

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

Alec Teal

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.

Ted Shifrin

2:40 AM

No one says dir der has to be continuous in $v$

Alec Teal

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

Ted Shifrin

That's cont except at $\pi/2$ ...

But the function is $0$ at $\pi/2$, I mean.

Alec Teal

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)

Ted Shifrin

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

Alec Teal

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!

Ted Shifrin

2:43 AM

Write it out carefully.

Alec Teal

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

Ted Shifrin

No, the limit.

When $\theta=\pi/2$ explicitly.

Alec Teal

@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 $\frac{0}{0+t^4}$

Ted Shifrin

Just put in $x=0$, $y=t$ ...

Right. Done. Your other algebra assumed ........

Alec Teal

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.

Ted Shifrin

2:51 AM

You tacitly assumed $\cos\theta\ne 0$ in your earlier alg.

Look at it after sleep.

Alec Teal

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?

Ted Shifrin

Except it isn't :)

Night! :D

Alec Teal

@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!

Studentmath

9:42 AM

It's official, I hate coding.

Balarka Sen

10:26 AM

@Studentmath What are you coding?

Studentmath

The stupidest things, as usual.

Balarka Sen

Like?

Studentmath

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

Balarka Sen

Oh. I thought you were coding math.

In that case, I can't help.

Studentmath

That wouldn't have been so terrible..

Balarka Sen

10:28 AM

@Studentmath Says who?

Studentmath

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

Balarka Sen

Yeah, that's kinda true.

Coding math is quite fun.

Why are you doing Java anyway, @Studentmath?

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..

Balarka Sen

Can't you just sign out?

Studentmath

The final test is within a month

Balarka Sen

10:31 AM

Oh noes. What're you going to do then?

Do they require Java for majors?

Not much zombies today, eh, @Studentmath?

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..

Balarka Sen

I am learning sieve theory. It's weird.

Studentmath

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

Balarka Sen

@Studentmath Interestingly weird.

Weird trick theory everywhere. Tricky analytic manipulations.

Studentmath

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

Balarka Sen

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.

Chris's sis

Greetings

Balarka Sen

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

Studentmath

11:12 AM

Reading!

Sounds extremely interesting. Any good reading sites/books?

@Balarka

Balarka Sen

11:27 AM

Sure. Try Halberstam's books.

Or, just download some survey on Brun sieve.

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.

Chris's sis

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

robjohn

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

Chris's sis

@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)

robjohn

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

Chris's sis

11:42 AM

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

Balarka Sen

12:42 PM

@Chris'ssis No need to write in nested form.

$\phi$ is much nicer.

Chris's sis

@BalarkaSen The nested form was written on purpose.

Balarka Sen

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

Chris's sis

@BalarkaSen Partial fractions? How?

Balarka Sen

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

Chris's sis

OK :-)

Balarka Sen

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.

Sush

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

r9m

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

Sush

Where is my reasoning wrong?

@r9m, please help!

r9m

@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

Sush

@r9m, Answer please!

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

Balarka Sen

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.

Chris's sis

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 ...

Parth Kohli

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

Chris's sis

2:01 PM

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

Jasper Loy

@Chris'ssis Can he resist your beauty then?

Balarka Sen

@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.

Chris's sis

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

Balarka Sen

@Chris'ssis "arrange"?

Chris's sis

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

Balarka Sen

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's sis

@BalarkaSen This one doesn't help you here ...

Balarka Sen

@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.

Chris's sis

@BalarkaSen no, no no ...

Balarka Sen

?

Chris's sis

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

Balarka Sen

2:18 PM

Hell.

I realize that.

Well, when there's a problem there always is a solution.

Chris's sis

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

Balarka Sen

$$\frac{1}{t(t-1)(\alpha t - 1)} = \frac{\beta}{(t-1)(t-\beta)} - \frac{\beta}{t(t-\beta)}$$

Now you do that.

Chris's sis

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

Balarka Sen

$t - 1 \mapsto t$

Jasper Loy

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

Balarka Sen

2:23 PM

Seesh, why does everything gets so complicated all the time.

Jasper Loy

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

Balarka Sen

@Chris'ssis No, that one is easy.

Chris's sis

@BalarkaSen Is it?

Balarka Sen

Use the zeta-gamma formula above

Chris's sis

@BalarkaSen Does it work? How?

Balarka Sen

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 $t/\beta \mapsto t$

Chris's sis

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

Balarka Sen

@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.

Chris's sis

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

Alyosha

2:42 PM

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

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$Mathematics$ Mathematics