Mathematics - 2014-08-26 (page 2 of 3)

Zachiel

16:03

I'm mostly looking for something less specific I'd be able to ask on the main site, or procedures that might get part of my problem solved and my original question smaller in scope.

So if anybody wants to hear the problem and see if s/he can help me unravel it a bit, just ping me.

leo

If the problem can be isolated, for sure someone would help

Zachiel

my problem as of now is isolating the problem. Let's see if I can explain it to someone who does not know the mechanics of the game.

Chris's sis

Oooo, what a nice discovery I did!

Zachiel

I suppose most of you are familiar with the concept of levels in a game. So, this character gains one "skillpoint" for every two "Intelligence (Int)" he has (rounded down) when he levels up. My aim is maximizing this skillpoint gain over several levels, by identifying the best way to spend resources on increasing Int.

so for example if he had Int=1 he would gain no points. Int=2 gains him one, as does Int=3

I have mainly three ways to increase intelligence.
One is starting from 0 or from 1 (with 1 being of course more favorable, but requiring me to make worse choices somewhere else in the system, so I'd be willing to analyse both cases separately)
The second one (and the one I want to focus on) is to buy some Int from a shop. Buying 1 int costs 5 times less than buying 5 int (the maximum I can get from this shop), but every time I buy I must pay again what I have already bought. Money is related to levels, and this is where the optimization problem lies. Examples in the next post.

Example: if I buy 1 Int now I pay it 100. If I buy another 1 Int later I pay it 200 (total: 300). Had I bought 2 Int in one chunk I would only have payed 200.

@Chris'ssis I don't know you, but - what did you discover?

@leo Let me know if the problem is clear enough to you.

Chris's sis

16:26

@Zachiel Do you really wanna know? Then I have some to write up.

Nick

Olla, guys. Any ideas on how to graph $x\cdot f(x)$ ?

Zachiel

@Chris'ssis If it's not something too complicated for me to understand given I'm not a math expert, and it's generally ok for this thing to be written on the chat, yeah, I'm curious. And you look like you want to tell that to someone, so... go on, please.

rehband

@Chris'ssis Show us! :)

Alexander Gruber

(never mind, i just can't read it)

Zachiel

:17353219 YEah, I did, but it looks like you missed one. My monetary income is low at the beginning and at every level I'm expected to gain more. Which means I might need to wait level 14 to buy 5 in a chunk, when (starting with Int=1) I could baybe buy 3 at level 10 and then buy 5 at level 15 and have a net skillpoint gain (at the expense of total money spent, which I don't really care about since it's gonna be pochet change by level, say, 33)

@AlexanderGruber putting it in more formal terms would be helpful, I guess. I've always been particularly bad with using the standard math lexicon.

Chris's sis

16:33

OK

Ted Shifrin

@Alex !!

hi @Nick, @Chris'ssis

Chris's sis

@TedShifrin Hello! :-)

Nick

@TedShifrin: Hi :D You wouldn't by chance be able to help me out with this:

8 mins ago, by Nick

Olla, guys. Any ideas on how to graph $x\cdot f(x)$ ?

Ted Shifrin

What do you mean, @Nick? What is $f$, and what do you mean by graph?

Zachiel

(uh, also, I really simplified the problem removing unnecessary concepts and using example numbers for the sake of simplicity. If we ever get to numbers I might need you to tell you exactly what numbers are on the level/gold matrix and what's the exact cost of a Int point at the shop)
Also, I just realized that probably starting from 1 and adding the extra 1 right at the first level is the optimum and I'll need to make the problem into a different one where I force Int(0)=0

Nick

16:38

@TedShifrin: ... f is real function probably trigonometric and by graphing, I mean plotting it on the Cartesian plane.

Ted Shifrin

Are we using only precalculus or are we using calculus? It would help to know what $f(x)$ actually is. For example, $x\sin x$ has a nice description. $xe^x$ is harder to see graphically without calculus.

Nick

@TedShifrin: In my actual question, $f(x) = |\sin x|$ but I was able to graph it... mostly. I couldn't get the amplitudes of all the waves though

mathh

If $q$ is a prime divisor of $2^{2^n}+1$, then why is it that $q\equiv 1 \pmod {2^{n+1}}$?

Ted Shifrin

There is no "amplitude", as $|x|$ is governing the high and low points.

user60887

if I have a function say x^2defined on [0,1] but I want to integrate broken points such as x\in{1/2, 1/4,1/8,1/6,1/32,....}? Is it even possible?

Nick

16:42

@TedShifrin: Yeah, the heights of the crests and troughs. XD

Ted Shifrin

What do you mean, @user6088?

@Pedro !!!

Nick

@PedroTamaroff: Olla :D

Ted Shifrin

@Nick, you can't answer that precisely. Even with calculus, it's hard to find the exact points of tangency. Those are transcendental equations.

What are you expected to do?

mathh

Could anyone help me?

If $q$ is a prime divisor of $a=2^{2^n}+1$, then why is it that $q\equiv 1 \pmod {2^{n+1}}$? I know that $a\equiv 1\pmod {2^{n+1}}$, but why is $q\equiv 1 \pmod {2^{n+1}}$?

Pedro Tamaroff

@Nick Are you cooking?

Nick

16:44

@TedShifrin: Just find the points of differentiability. I did it. No probs. When I graphed the function, the high and low points disturbed me.

user60887

meaning if f(x)=x^2 over [0,1] but x are points coming from the set {1/2,1/4,,1/8,1/16,... is it possible to integrate this?

Nick

Pedro Tamaroff

@TedShifrin Hello.

Nick

@PedroTamaroff: ... depends on your definition of cooking

Ted Shifrin

The high and low points are no longer at the multiples of $\pi/2$, @Nick. Close, but not there.

You recovered and back to work, @Pedro?

I still do not know what you mean, @user6088? Are you trying to evaluate some sort of Riemann sum?

user60887

16:46

yes

Nick

@TedShifrin: So, I have to go all Newton Raphson to find those points. :0

Ted Shifrin

Yup @Nick

Sab ಠ_ಠ

Hello everybody :)

Nick

I'd rather go screw an integral

Sab ಠ_ಠ

Hello @TedShifrin :)

Ted Shifrin

16:47

slaps @Nick for impudence

salut @Sab

Mike Miller

@Pedro Are you home?

Sab ಠ_ಠ

Quoi'd neuf?

Ted Shifrin

He's probably on his phone @Mike

Pedro Tamaroff

@MikeMiller Yes.

Nick

@TedShifrin: Thanks, I needed that

Sab ಠ_ಠ

16:48

I'm getting a brand new Spivak for myself :3 :3

It was the last in the country :3

Nick

@Sabಠ_ಠ : Which country? It better not be mine.

Sab ಠ_ಠ

Now I can return the one from the library :3

Ted Shifrin

Félicitations, @Sab. Now make sure you can do 2/3 of the problems in it :)

Sab ಠ_ಠ

S.A

Pedro Tamaroff

@TedShifrin I'm trying to finish a series before my Netflix free month expires! How's that for a challenge...

Alexander Gruber

16:49

@TedShifrin I went to a diffgeo class the other day.

Ted Shifrin

LOL @Pedro ... Life is tough. I thought you were back in classes ... having missed a week? :P

Sab ಠ_ಠ

@TedShifrin it's a book I always wanted to own :) I'm pretty sure it will help even in higher classes

Ted Shifrin

Ah, @Alex, right. How was it?

Alexander Gruber

but i may not be able to keep taking it. It's at the same time as another thing.

Mike Miller

@Ted Home in the sense of Buenos Aires.

Ted Shifrin

16:49

The US deported him, @Mike :)

Pedro Tamaroff

@TedShifrin I just got back from ComplEx Analysis.

Alexander Gruber

@TedShifrin it looks real good. real good.

do you know Prof Robinson from UF?

Sab ಠ_ಠ

@TedShifrin remember when I asked if it's possible to complete in 2 months?

Ted Shifrin

Cool, @Alex. No, but you showed me a syllabus from David Greisser, I thought.

yup @Sab.

Sab ಠ_ಠ

I'm thinking of using Apostol, because of more content.

Mike Miller

16:50

@TedS Took long enough!

Sab ಠ_ಠ

But is it doable? Just Apostol?

Alexander Gruber

@TedShifrin yeah, the course changed hands i think.

Ted Shifrin

No, @Sab, not even close.

Sab ಠ_ಠ

:S

Aie!

How would you suggest I go about it?

2 months before exams

Ted Shifrin

Aren't you taking an actual course, @Sab?

Sab ಠ_ಠ

16:51

I am

Ted Shifrin

So you need to learn in that framework. Doing Spivak or Apostol is only for extra enlightenment.

Sab ಠ_ಠ

But its not enough. I don't feel I learned much.

Aha

Ted Shifrin

Well, first priority is to learn everything 100% in your course. Then add to it.

Nick

@Sabಠ_ಠ : Stick onto that feeling!

Sab ಠ_ಠ

Okay

Ted Shifrin

16:52

@Pedro: How far is the complex analysis course now?

Sab ಠ_ಠ

@Nick I will :)

Nick

@Ted: No one uses impudence anymore!

Pedro Tamaroff

@TedShifrin meh/10

Ted Shifrin

Are you calling me arcane and archaic, @Nick?

Nick

Graph shows word usage with respect to years

@Ted: As long as they don't imply anything bad, yes!

Ted Shifrin

16:54

How exciting @Pedro.

Pedro Tamaroff

@TedShifrin Exactly.

Ted Shifrin

Complex analysis has always been one of my favorite courses to take/teach.

Maybe I should send you a few problems, @Pedro :) Actually, the book you stole from me :) has a number of great ones.

N3buchadnezzar

@TedShifrin Which book? =)

Ted Shifrin

Polya-Szego Problems in Analysis, @N3b

N3buchadnezzar

*scratches head

Chris's sis

16:58

Back

@Zachiel @rehband

Prove that

$$\lim_{n\to\infty}\left(\frac{3}{2}-2\log(2)\right)-\left(-\frac{1}{2}-\eta(2)+2\log(2)\right)+\left(\frac{3}{2}+\eta(2)-\eta(3)-2\log(2)\right)-\left(-\frac{1}{2}-\eta(2)+\eta(3)-\eta(4)+2\log(2)\right)$$ $$+\cdots+(-1)^{n+1}\left(\frac{1}{2}(1-2(-1)^{n})-\eta(n)+\eta(n-1)-\eta(n-2)+\cdots+(-1)^{n+1} (\eta(2)-2\log(2))\right)=\frac{3}{4}-\eta(2)$$

Alexander Gruber

does anybody have a convoluted proof that $\sum x^i=\frac{1}{x-1}$?

Pedro Tamaroff

@TedShifrin Yes, true. I will look at them eventually. I have to prepare final exams first!

Sab ಠ_ಠ

I'll see you guys later. I'll go do some probability theory. :)

See y'all

Zachiel

@Chris'ssis Ok, that's too complex for me. I don't even understand approximately half of the notation.

Is there a script that should let me see that as a formula?

Ted Shifrin

Oh right, @Pedro. I forgot that after your birthday holiday came exams.

Khallil

17:01

What do you mean by convoluted, @Alexander?

Alexander Gruber

@Khallil excessive

Ted Shifrin

@Zachiel: See "LaTeX in chat" to the right

@Alex, is the back doing any better?

Alexander Gruber

like, using category theory, or something.

Pedro Tamaroff

@TedShifrin I will try to sit for Calculus and Algebra II mainly... and Analisis II.

Alexander Gruber

@TedShifrin got a neurology appt tomorrow, gonna finally get to the bottom of things i hope

Khallil

17:02

Oh. Not a clue. The only one I use is to do with geometric series, @Alexander.

Chris's sis

I can rewrite that in a nicer form.

Ted Shifrin

I sure hope so, @Alex. You need some pro ...

Alexander Gruber

Yeah headaches and neck pain every day are not real condusive to a career as a mathematician

Ted Shifrin

Sigh, @Alex. I wish there were something I could do to help. Rooting strongly for you.

Alexander Gruber

thanks, i do appreciate the concern

Khallil

17:04

I know somebody whose (is that the correct 'whose'?) neck pain was attributed to bad posture, @Alexander.
Lets hope it's all ok tomorrow!

Ted Shifrin

Yes, whose is correct, @Khallil. But a horrid accident is the cause for Alex. :(

Alexander Gruber

@Khallil Pretty sure mine's from the car that ran into me. :P

but, i do not take posture lightly in general

Khallil

Posture's harder to correct as you get older. Ah, it's just one of those things!

Zachiel

@TedShifrin thanks

Alexander Gruber

@Khallil i been working on correcting an anterior pelvic tilt for a little while and that's been going really well

Jasper Loy

17:09

Wow this chat is alive!

Khallil

Alive and kickin', @JasperLoy!

Jasper Loy

@AlexanderGruber You misspelled conducive.

Ted Shifrin

No, @Jasper, some of us are unalive.

Alexander Gruber

@JasperLoy That's the Kentucky spelling.

it's kind of like how british words are spelled differently. Very acceptable.

Jasper Loy

@Khallil You mean Thomas Andrews, lol.

Why all removed?

rehband

17:14

@Chris'ssis Wow, crazy :D Nice work. What's $\eta$?

Chris's sis

Prove that

$$\lim_{n\to\infty}\left(\frac{3}{2}-2\log(2)\right)-\left(-\frac{1}{2}-\eta(2)+2\log(2)\right)+\left(\frac{3}{2}+\eta(2)-\eta(3)-2\log(2)\right)-\left(-\frac{1}{2}-\eta(2)+\eta(3)-\eta(4)+2\log(2)\right)$$ $$+\cdots+
\left(\frac{1}{2}((-1)^{n+1}+2)-2\log(2)+ \eta(2)-\eta(3)+\cdots+(-1)^{n+1}\eta(n)\right)=\frac{3}{4}-\eta(2)$$

Dirichlet eta function

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). The following simple relation holds: While the Dirichlet series expansion for the eta function is convergent only for any complex number s with real part > 0, it is...

Sawarnik

@khallil What is this removed stuff? :O

@nick Hey.

Chris's sis

I can still arrange things there. I'll do this soon.

Zachiel

Anyway, for my question earlier, if you were to formalize it for asking it as a question on the main site, where would you start?

Worst editing skills ever...

Uh no I was talking about me editing "my wuestion" into "myquestion"

However, yeah, Sum form 1 to infinite is better than a limit in this case since it appears to be a series. Assuming I translated all the things correctly from my language.

Chris's sis

Something is wrong in my formula.

It should be $$\sum_{n=1}^{\infty}\left(\frac{1}{2}((-1)^{n+1}+2)-2\log(2)+ \eta(2)-\eta(3)+\cdots+(-1)^{n}\eta(n)\right)=\frac{3}{4}-\eta(2)$$ I'm in a hurry now, but I'll check that again when I'm back and post a final version.

r9m

17:28

@Chris'ssis Hello :) .. I had many classes today .. here & there =) finally back to my nest =P

rehband

@Chris'ssis Ah ok]

Chris's sis

@r9m Hi. Sorry I have to go now. Back in half an hour or so.

r9m

@Chris'ssis ;) okay :))

Chris's sis

Can someone check that numerically?

Out for now

Zachiel

Since there's been quite a change of logged-in people here: I wonder if my problem is fit for a question on this site and wheter it is or not I'm looking for farmalizing the problem or help in deciding how to start solving it.
Its brief discussion starts here: http://chat.stackexchange.com/transcript/message/17352945#17352945

and bye @Chris'ssis

Alec Teal

17:36

@TedShifrin I'm looking for a book (some books/notes) on Singularities I've found 4 books, 2 are not concerning what I think they are, wondering if you can help.

@Zachiel that's a combinatorial optimisation problem

Arguably "Utility" of decisions

Zachiel

@AlecTeal yes, I think it is... I also guess I need to ask other people who actually play the game if some of the options I'm considering are actually viable or if I can reduce the scope of the problem, like by setting initial Int to 0

Ted Shifrin

@Alec: Depends what you're looking for.

Alec Teal

@TedShifrin what are my options, one of the books is a 2 volume thing from a symposium, filled with papers (One written by you! - Edited by my tutor, small world!) - I'm not having much luck finding anything else.

leo

@Zachiel So, every time the character levels up, his skill points (spts) are recalculated by $$\mathrm{spts} = \left\lfloor \frac{\mathrm{Int}}{2}\right\rfloor.$$

Zachiel

17:52

not recalculated. They're added to a running total.

leo

so

$$\mathrm{spts} = \mathrm{spts}+ \left\lfloor \frac{\mathrm{Int}}{2}\right\rfloor.$$

Zachiel

exactly. And every time you add to the total you need to consider the current Int

leo

then we should always minimize the resources spent in gaining Int

Zachiel

which means that increasing Int early in levels is the best option, was it not that you need to pay

I'm going to dine now, I'll answer you later.

Jasper Loy

@AlecTeal What kind of singularities?

Alec Teal

17:57

Not sure, not the stationary load balancing one

leo

@Zachiel A little observation: one should always aim for an even amount of Int, because if you have $n$ even, $\mathrm{Int} = n$ and $\mathrm{Int} = n+1$ give the same $\mathrm{spts}$ and of course $\mathrm{Int} = n$ is cheaper

Jasper Loy

@AlecTeal Not sure? Then how can we help you?

Alec Teal

Well @TedShifrin has written about it. So he knows. Also my struggle is why I am asking.

Jasper Loy

18:15

Wow, I just learnt that we have a blog, lol.

Mike Miller

@TedShifrin GP's arguments for oriente intersection theory all rely on taking the orthogonal complement in tangent spaces regularly. I can salvage this in general just by picking my favorite metric, right?

Khallil

The removed messages are nothing to worry about, @Sawarnik. ^_^

mixedmath

@JasperLoy 'tis young, and uncertain

Khallil

What was I doing yesterday?
http://chat.stackexchange.com/transcript/message/17337803#17337803

Alec Teal

GTG catch a train now!

Buggers keep getting away so wish me luck!

mixedmath

18:24

@AlecTeal run run!

Mike Miller

Please stop spamming...

@mixedmath You intend to write something, right? I'd forgotten what.

Alizter

@Chris'ssis I need integrals

skullpatrol

You are really asking for it pal :-)

Zachiel

@leo Of course. But that could be done with starting from 0 and buying 2, then buying 4, then buying 5 and at the same time applying the free increase; or by buying 1 and the increase, then 3, then 5; or by buying 2, then 3 and the free increase, then 5, Or any of those but without one of the steps (bar the last, which is mandatory)

mixedmath

@MikeMiller I actually have several things in mind, and I have one written. The three I'm most interested in are 1: Zipf's law analyzed on MSE's data dump (I've done the analysis, but I haven't made pretty pictures - those take a while). 2: Sort of an expository look at probability in popular media, like what a 20% chance of rain really means. 3: How to find asymptotics for the number of integer points on one-sheeted hyperboloids with modular forms

The one I have written is number 2, but I've been putting some real thought into pop math recently, and I'm planning on giving it a serious edit through

robjohn

18:38

@mixedmath I have some images for Benford's Law in a post... I don't know if they will be useful, but let me find them

mixedmath

@robjohn oh, interesting

robjohn

@mixedmath here

Mike Miller

@mixedmath Those all sound interesting! I know very little about pop math - does it usually get things as "wrong" as pop science does?

mixedmath

@robjohn that's a nice post, and a visually appealing graphic. I also want visually appealing graphics - but they're hard (at least to me)

@MikeMiller usually, it does. Or it's written for someone who only confesses to know math at the 5th grade level, so anything above arithmetic algebra is handwavy and often, therefore, "wrong"

I'd say that a lot of Gardner's work was pop math, and it wasn't handwavy at all. Ellenberg just put out a book called "How not to be wrong", which is pretty unhandwavy too.

actually, I find Ellenberg's writing to be very engaging

reading pop math (and pop science, for that matter) is a mild obsession of mine

Mike Miller

@mixedmath how are Stewart's books?

robjohn

18:45

@MikeMiller The one thing that I think misleads a lot of people is things like regularized summation as in this answer.

Mike Miller

That's a more recent confusion, I think... I'd never really heard widespread talk about it until the numberphile video.

Chris's sis

The corrected version $$\sum_{n=1}^{\infty}\left(\frac{1}{2}((-1)^{n+1}+2)-2\log(2)+ \eta(2)-\eta(3)+\cdots+(-1)^{n}\eta(n)\right)=\eta(2)-\frac{3}{4}$$

What can I say now?

leo

@Zachiel so one does have to apply the free increase at some point before one finish the current level? Also, one can go an unlimited number of times to the store per level or is there a restriction?

Chris's sis

$$ \eta(2)-\eta(3)+\cdots+(-1)^{n}\eta(n)=\mathcal{O}(n) $$

robjohn

@Chris'ssis $\eta$ being the alternating $\zeta$?

Chris's sis

18:50

@robjohn Yes.

Mike Miller

@robjohn I must admit I get an upset stomach looking at Lubos Motl's answer

Chris's sis

@robjohn Did you know the asymptotic above?

robjohn

@Chris'ssis If you remove the infinite summation, yes :-)

Chris's sis

Wait, not that way !!!!!!!!!!

@robjohn My notations fail once in a while ... :-(

(and I'm pretty tired now)

mixedmath

@MikeMiller I like that he isn't afraid to write down some real math. He still tries to hit a really low common denominator audience, but some of his works are really nice. His book "From Here to Infinity" was really good. I thought Letters to a Young Mathematician was not. I read Flatterland in high school, and I liked it then - but I don't think it held up too well.

robjohn

18:55

@Chris'ssis is that better? or were you saying the infinite sum converged to something nice?

Chris's sis

@robjohn Yeah, I was referring to the infinite sum, but let that there until I post other stuff.

robjohn

@Chris'ssis Ah, I would believe the sum to $n$ if the end term were $(-1)^k\eta(k)$

Chris's sis

@robjohn Exactly. Anyway, I'll attend this part again. Thank you for removing that crap above.

Zachiel

the free increase happens whenever I want, basically. In the game, it's the effect of me aging enough to became wiser. If it happens while I'm level 13, I will benefit from it starting with the 13 to 14 levelup, when spts are added to the running total for the first time since.
As for going to the shop twice during a level, what for? If I buy 1Int and later during the same level 2Int (without using that 1Int in any way meanwhile since I didn't level up) i'd better have bought 2Int in the first place.

robjohn

@Chris'ssis I can correct it, or remove it. Which would you prefer?

Chris's sis

19:00

@robjohn Let things the way you did them. No need for any other action. It's OK.

:-)

robjohn

@Chris'ssis certainly

Zachiel

@leo But maybe I see where you're coming from. Let's suppose at every level up I first add up my spts, then gain a progressively bigger abount of money that's not really tied to any mathematical progression and which I wont publish here for copyright reasons.

Chris's sis

@robjohn I'm referring to the asymptotic of this sum $$\sum_{k=2}^{n} (\eta(2)-\eta(3)+\cdots+(-1)^{k}\eta(k))$$ This is what I wanted to ask you above, if you know this asymptotic.

Mike Miller

@mixedmath I have no experience with his pop math books. But he sounds better than most.

Zachiel

I also understand I might limit my analysis on the range of levels between the one needed to buy 1Int and the one needed to buy 1Int+2Int+3Int+4Int+5Int

Chris's sis

19:06

@robjohn if looking at my limit above, you easily figure out the asymptotic.

anorton

Yay! I finally was able to use something from Concrete Mathematics to solve a problem here! (summation via finite calculus)

Sorry for bursting in, but I was too excited not to say something...

leo

@Zachiel Another question, every time one goes to store, the value per Int point doubles, right?

Tunk-Fey

Could anyone here check my answer (math.stackexchange.com/a/910148/123277) numerically using software. I don't have any and I am worried about my answer gets downvoted.

Chris's sis

@robjohn my point is that $$\lim_{n\to\infty} \frac{1}{(2\log(2)-1)n}\sum_{k=2}^{n} (\eta(2)-\eta(3)+\cdots+(-1)^{k}\eta(k))=1$$ :D

anorton

@Tunk-Fey Oh my goodness!! That's amazing...

(I don't have mathematica, so I can't check it. But your answer represents an immense amount of effort...)

leo

19:15

@Zachiel If we can calculate $\operatorname{bs}(n)$ for each $n$, where is the best possible spending in $n$ Int points, that's the lowest we can manage to pay for $n$ Int points, we can forget of how the money increases

@Zachiel So we can assume that the total Int points will be increased at least by 1 always, at the end of each level

Tunk-Fey

@anorton Thanks but my answer is different with Cleo's. I think I've made a mistake but I don't know where.

anorton

@Tunk-Fey Your numeric result doesn't match Cleo or Mhenni Benghorbal's. (I plugged in your final answer to Wolfram)

Mike Miller

@Tunk-Fey I can check in 5-ish hours computationally, if nobody else does.

Tunk-Fey

@anorton I know that, even WA couldn't help me about its numerical value.

@MikeMiller Thanks. it's 3.30 am here in Indonesia, I want to take rest.

5space

19:31

@MikeMiller, question about something in Lee.

Chris's sis

Let me change the statement: Prove without pen and paper that $$\lim_{n\to\infty} \frac{1}{(2\log(2)-1)n}\sum_{k=2}^{n} (\eta(2)-\eta(3)+\cdots+(-1)^{k}\eta(k))=1$$

Mike Miller

@5space hello

Chris's sis

@r9m Did you see my last creation? $$\sum_{n=1}^{\infty}\left(\frac{1}{2}((-1)^{n+1}+2)-2\log(2)+ \eta(2)-\eta(3)+\cdots+(-1)^{n}\eta(n)\right)=\eta(2)-\frac{3}{4}$$

r9m

@Chris'ssis ya :)) !! .. atm 'm so tired .. I feel like gazing at stars :P lol .. I'll try and think about it after some sleep :P

5space

@MikeMiller, If $G = N\rtimes H$, $\tilde N = N\times\{e\}$, and $\tilde H = H\times\{e\}$, am I supposed to interpret $\tilde G$ to be $\tilde N \rtimes \tilde H$?

Chris's sis

19:38

@r9m I only wanted to show them to you. :-)

r9m

@Chris'ssis (y) .. :-)

Mike Miller

@5space Why the hell is this in Lee?

5space

Lie groups

Mike Miller

What does \rtimes denote? Surely not semidirect product.

5space

Yes

Mike Miller

19:43

Oh, just a different stupid group structure on the product manifold

Pedro Tamaroff

@MikeMiller Booo Mike, boooo.

Mike Miller

I don't remember this at all.

5space

$N$ and $H$ are Lie groups, $\theta$ is a smooth action of $H$ on $N$ with $G=N\rtimes_\theta H$.

Mike Miller

I guess it's still smooth because the automorphisms in question are Lie group isomorphisms, yeah.

5space

I was just trying to figure out if $\tilde G$ was the semidirect product of $\tilde N$ and $\tilde H$, because he didn't explicitly say so.

Mike Miller

19:44

Sure, your thing sounds reasonable, I guess. What's the context?

5space

Show that $\tilde N$ is normal subgroup of $\tilde G$.

r9m

@Chris'ssis how do you explore such bizarre series ?!! they come up in context of your research .. but as a viewer of your select gems .. I always have very little clue about the big picture .. :]

Mike Miller

It doesn't really make sense to me. Why would \tilde G be different than G?

5space

I don't know. I'm confused :-(

Mike Miller

Is that literally all it says?

Chris's sis

19:46

@r9m They all naturally come in my research. It's nothing special, just some study of the things. :-)

Mike Miller

What's the problem number/page? I'm just going to look at it in SEL lol

5space

Yeah, it defines $G,N,H,\tilde N,\tilde H$, asks you to show that $\tilde N$ and $\tilde H$ are closed Lie subgroups of $G$ isomorphic to $N$ and $H$ (resp) and then asks to show that $\tilde N$ is normal subgroup of $\tilde G$.

Maybe it is a typo and $\tilde G$ was supposed to be $G$?

r9m

@Chris'ssis 'kay :-)

5space

I don't see it in the corrections...

but it's for the 2nd edition

oh nvm I thought it was in 3rd edition now :-P

Mike Miller

Lol you checked it out didn't you

5space

19:50

yes

mwahahah

It seems to me that $\tilde G$ is a typo and should be $G$...

Mike Miller

Makes sense to me. Though in that case the problem is essentially trivial.

5space

But I don't know what $\tilde G$ would even mean?

I mean it's just an exercise in the text (not a problem), so it very well could be trivial

Mike Miller

Just assume it means $G$, it's the only thing that makes sense to me.

5space

Coolio, thanks :-)

Mike Miller

Now kill the problem in cold blood.

5space

19:54

I'm trying to finish ch7 and get started on ch8 :-3

mixedmath

@5space which book?

5space

And I'll hit Ch9 at least, then!

@mixedmath: Lee's smooth manifolds

Mike Miller

Chapter 9 can go to heck

5space

That's how I feel about Ch 6

Mike Miller

Sard's and stuff?

5space

19:57

Yeah.

Mike Miller

It's technical but the stuff that's not Sard's theorem itself is enlightening

I have not yet benefited from knowing the proof of Sard's theorem.

mixedmath

@5space ah, that's what I learned from too. Good days. I'm tutoring a very advanced student right now who wants to look at Lee's Smooth Manifolds.

5space

I was told to read most of the chapter sans proofs :-P

mixedmath

I'm a bit worried - I've forgotten a lot

5space

It's a cool book! You could pick it up again while you're tutoring, I'm sure :-)

Mike Miller

20:00

@mixedmath Fair warning: it's structured much differently than the first edition

Alizter

@Chris'ssis $\displaystyle \int_0^1\frac{x^n}{\log x}\mathrm dx=\gamma+\log(n+1)$

Chris's sis

@Alizter hmmm, are you sure everything is OK there?

Mike Miller

But you should pick it up again pretty easily... there aren't many ideas in each chapter, so in broad strokes, it's pretty simple

Alizter

@Chris'ssis Why?

Positive ns anyways

Chris's sis

@Alizter Check the case $n=1$

mixedmath

20:05

@MikeMiller oh, interesting

Alizter

@Chris'ssis Show me the problem.

Chris's sis

@Alizter How do you integrate $x/\log(x)$? And how do you get those results ...?

Alizter

@Chris'ssis start with $\log x = t$

Chris's sis

According tou your result, we have that $$\displaystyle \int_0^1\frac{x}{\log x}\mathrm dx=\gamma+\log(2)$$? Check it some more.

5space

@MikeMiller, I emailed it to Lee and he just emailed back saying it was a typo and added it to the corrections.

Too bad it isn't Knuth so I could get some $$$

Zachiel

20:10

Not really. They sell magic books that set your "book bonus" to a number. Each book costs proportionately to that number, but since they set the number rather than adding to it... buying a 5Int book after having bought a 4Int one costs as much as buying it as your first and only book.
I don't think I really understand that bs(n)... maybe we can have a chat where we make some examples, not to flood here? I'd have it on RPG.SE if it's ok for you. Because no, we can't make that assumption. (and sorry for being late)

Mike Miller

@5space not real $$$

Chris's sis

@Alizter Just look at the way the integrand behaves near $x=1$.

Alizter

@Chris'ssis I found the error

leo

@Zachiel would be better to play the game :-)

Alizter

I missed something which caused values for gamma to come up

5space

20:14

@MikeMiller, Yeah Knuth paid people who found typos in his books.

Alizter

cool but wrong

Chris's sis

@Alizter We know that $$\lim_{x\to1} \frac{\log(x)}{x-1}=1$$ and hence we see that the integrand behaves like $$\frac{x}{x-1}$$ in that region I mentioned.

leo

I have to go soon. But if you do the room, post there I'll check it later

Mike Miller

@5space Not anymore. They're now not real checks. :P

5space

Le sigh :-(

Alizter

20:15

@Chris'ssis I thought I had discovered something interesting for a second :P

Zachiel

@leo Yeah I know ;)

Chris's sis

@Alizter Well, keep working on things, discovering things, sometimes we fail, but the failing let us the possibility to learn lots of other things. :D

Zachiel

@leo chat.stackexchange.com/rooms/16720/mathematics-applied-to-dd here it is

I'll start by talking about the problem again, as an introduction for people who, like you, don't know the game.

and thanks for your help ;)

Alizter

20:32

@Chris'ssis I was doing stuff on group theory when it randomly came to me

Bob

Hii @Zachiel......

5space

@mixedmath, do you know Isaac?

Bob

@Chris'ssis.... can you please tell me when is the function same as the inverse FT of its FT

Mike Miller

@5space do you know Isaac?

5space

I never met him, because he graduated from UCLA the year before I came, but all of my friends know him.

@MikeMiller do you know Isaac?

Mike Miller

20:47

I knew you didn't know Isaac.

5space

I've talked to him on SE :-P

Mike Miller

@5space No idea who he is.

Bob

@mike miller can you please help

Chris's sis

@Alizter I see.

Alizter

@Chris'ssis so when you are stuck. do something else and the amazing thing called the human brain comes up with stuff :)

Chris's sis

20:53

@Alizter Exactly! :D

mixedmath

@5space yeah, he's a really great guy

one of the grads I'm closer to here (whatever that means)

5space

Nice, I never met him in person, but all of my friends speak very fondly of him.

(And he's super nice on SE!)

Mike Miller

21:05

@Bob Probably not, but even if I could, I don't look at problems pinged to me out of the blue.

Pedro Tamaroff

@MikeMiller Oh, no. Mike is offended.

Alizter

@MikeMiller My car won't start

@Chris'ssis $\displaystyle \int_1^2 \int _1^2 \frac{\psi(x)\psi(y)}{\psi(x+y)}\mathrm dx\ \mathrm dy$

I am trying if I can make some cool polygamma beta function

Bob

@MikeMiller when is the function same as the inverse FT of its FT

???

Mike Miller

21:24

@Pedro It's the principle. But it doesn't seem to have mattered.

robjohn

@Chris'ssis Since we have the following, I agree:

$$
\begin{align}
\sum_{k=2}^\infty(-1)^k\eta(k)
&=\sum_{k=2}^\infty(-1)^k\sum_{j=1}^\infty\frac{(-1)^{j-1}}{j^k}\\
&=\sum_{j=1}^\infty\frac{(-1)^{j-1}}{j(j+1)}\\
&=\left(1-\frac12\right)-\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)-\dots\\
&=1+2\left(-\frac12+\frac13-\frac14+\dots\right)\\
&=2\log(2)-1
\end{align}
$$

Jose Antonio

HI guys I was working in this exercise and I have problem to follow the answer of Hagen, someone could give more details, please?

1

Q: If $G$ is a simple group, $f:G \to H$ is a homomorphism, and $A\lhd H$ is such that $[H:A]=2$, show $f(G) \subset A$

I'm stuck with the following problem. Can someone help me by providing a hint? Suppose $G$ is simple and let $f$ be an homomorphism between $G \to H$. If $\#G\ne2$, $A\lhd H$, and $[H:A]=2$. Then $\text{Im}f$ is a subgroup of $A$. I know that the kernel of the homomorphism is a normal subgroup...

Chris's sis

@robjohn Great job! :-) After applying Stolz theorem, that is the only thing to compute.

(I barely manage to type --- some sleep is needed)

Ted Shifrin

@mike: You can restate it with quotient spaces. Transversality is equivalent to having the induced map $df_x\colon T_x X \to T_{f(x)}Y/T_{f(x)}Z$ be surjective. Or is this not what you're talking about?

@Alec: Who's your tutor? And what specifically are you trying to learn? I'm not even sure what symposium you're referring to. I've worked on several projects that involved singularities, and there's a huge variation in topics.

@Mike: Oh, are you talking about normal bundles? That's usually done with a metric, but, again, in algebraic worlds, one does quotient bundles.

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$Mathematics$ Mathematics