Mathematics - 2013-01-28 (page 2 of 3)

11:49 AM

We say that Infinity is defined, yes?

I know Infinity is not a number, but this is a different concept to not being defined.
How can I convince someone that these concepts are different?

UnkleRhaukus

12:10 PM

How is 'defined' defined?

3

UnkleRhaukus

12:36 PM

I guess it requires some kinda circular argument?

user58512

12:57 PM

@UnkleRhaukus, haha how is defined defined

robjohn

1:20 PM

@JonasTeuwen Good morning

user58512

@robjohn, i accepted your post

robjohn

@user58512 I saw. Thanks. I found out that my previous method didn't work out as I'd thought.

user58512

I didn't find a book about convergence in the library today, wil have to look again tomorow

robjohn

@user58512 The notation is just very hard to write down. Once the proper equations are shown, it is pretty simple.

user58512

@robjohn, Yeah, it's really hard to , you have to be so careful and everything is infinitely long - but it does work out perfectly.. like magic. any idea why Lambert thought to take continued fraction of sin(r)/cos(r)?

robjohn

1:25 PM

@user58512 There is probably another approach that makes things more evident.

user58512

do you have any thoughts on this? mathoverflow.net/questions/42386/…

I want to reask it but if it didn't get answered there it probably wont here..

even if you just take 10 terms of the continued fraction, evaluating it at pi/2 gives you something around -10^10

why is that?

robjohn

1:43 PM

@user58512 I don't see anything.

user58512

there

I asked it as a question

robjohn

$$\frac{\pi }{1-\frac{\pi ^2}{3-\frac{\pi ^2}{5-\frac{\pi ^2}{7-\frac{\pi ^2}{9-\frac{\pi
^2}{11-\frac{\pi ^2}{13-\frac{\pi ^2}{15-\frac{\pi ^2}{17-\frac{\pi ^2}{19-\frac{\pi
^2}{21-\pi ^2}}}}}}}}}}}=1.6857393410602451374\times10^{-9}$$

user58512

yeah

why does it go to zero near pi?

robjohn

Because $\tan(\pi)=0$

user58512

can we deduce thatfrom the continued fraction

I dont know what the relation is between tan and the continued fraction

robjohn

2:00 PM

@user58512 Isn't that the continued fraction we just worked out?

user19161

Jonas doesn't seem to be coming back to chat, sad panda. Well, I hope he returns soon!

robjohn

@JasonBourne Asaf followed the same pattern. Suspension then self-imposed exile.

user58512

All we've shown is that tan(r) = r/(1-r^2/(3-r^2/[a(r)/b(r)])) where a,b are some power series functions

for any finite depth

I don't know how to assign meaning to the continued fraction as an object in its own right

user19161

@robjohn I did not know what happened exactly a few hours ago, I may have misunderstood matters. Maybe Jonas didn't call anyone anything but there was a misreading, but never mind.

robjohn

@user58512 Read up on continued fractions. Usually as a function of the very bottom terms, they don't vary much.

user58512

2:04 PM

that's true for simple continued fractions

robjohn

I am used to working with simple continued fractions, but the same thing works with others.

user58512

eventually x will be dominated by the odd number in "odd - x^2/", so we have reason to believe this will work here too

@robjohn, check this out it is so funny cut-the-knot.org/wiki-math/…

robjohn

@user58512 play with the quantity subtracted from the 21 and see what happens.

user58512

if you didn't see it already

Frank Science

Is there anybody having read Gödel, Escher, Bach: An Eternal Golden Braid?

user58512

2:07 PM

@FrankScience, no, if you have a question about math from it maybe someone knows though

Frank Science

@user58512 I wonder whether it's worth buying.

user58512

I doubt it

robjohn

@user58512 $x=\frac{2}{3-x}\Rightarrow x^2-3x+2=0\Rightarrow x=1\text{ or }x=2$ cute

user58512

hmm

oh the thing I linked!

Orange Harvester

@JasonBourne Yo.

user19161

2:15 PM

@OrangeHarvester Yes?

Orange Harvester

@JasonBourne Just pinging hi.

user19161

@OrangeHarvester Oh OK. I get many emails from SE users these days, which is a good thing. =)

Orange Harvester

@JasonBourne =)

Hehe, one of the restaurant is providing me a loyalty bonus till 31st Jan. 20% off. Must grab it soon.

user19161

@OrangeHarvester That is like a few days left.

Orange Harvester

@JasonBourne Yes. Four days worth of 20% free.

user19161

2:18 PM

@OrangeHarvester Well, if you don't like the food don't bother. It's just a trick to waste your money.

Orange Harvester

@JasonBourne Well, food is awesome. And anyway, I am going to eat somewhere, might as well be there.

user19161

Haha, I just answered a lhf, nobody wants to upvote it...

Orange Harvester

That's the way to go!

link?

user19161

math.stackexchange.com/questions/288921/… Here it is. Well, I guess we need to wait longer for votes to come in.

user19161

Some comments should be answers, and some are totally misleading.

Orange Harvester

2:22 PM

Ahh, I know squat about logic. :-)

user19161

Yes, I think so too. =)

user19161

But that question is just really trivial stuff.

robjohn

@user58512 $$P_n(x)=\sum_{k=0}^\infty(-x^2)^k\dfrac{2^n(k+n)!/k!}{(2k+2n+1)!}=\left(\frac1x\frac{\mathrm{d}}{\mathrm{d}‌x}\right)^n\frac{\sin(x)}{x}$$

just for fun

user58512

cool!

robjohn

off to the park

Nimza

2:40 PM

Hello

Help me please, what is the easiest way to show that if $\deg f = n$ and $f$ is a local homeomorphism $S^1 \to S^1$ then $\mathop{\mathrm{card}} f^{-1}(x) = |n|$ for any $x$?

Greg Ros

2:53 PM

Is there an example of a function that is defined everywhere, but is bounded nowhere (by which I mean, in no interval)?

Tobias

@GregRos hmm, I think any non-continuous solution to f(a + b) = f(a) + f(b)

as any non-empty open set in $\mathbb{R}^2$ intersects the graph of such a function in a non-empty set

Frank Science

@GregRos Canway base 13 function is elementary.

Orange Harvester

@GregRos What do you mean, when you say defined? For example, if we consider $\tan x$ to be defined at $\pi/2$, then one of the solutions can be inverse of an indicator function over rationals.

Frank Science

@GregRos Another example here: $f(x)=n$ where $x=m/n, n>0$ and $\gcd(m,n)=1$, otherwise, $f(x)=0$.

Tobias

@OrangeHarvester how can we consider tan defined there?

@FrankScience nice example

Orange Harvester

3:00 PM

@Tobias We cannot, but then I am wondering, if a function is defined everywhere, it has to be finite, and hence bounded. (I might be missing something.)

Tobias

@OrangeHarvester see the examples just given

Orange Harvester

@Tobias Yes. That solves my doubt.

@FrankScience Nice example.

Tobias

for a continuous function, we cannot do this of course

Greg Ros

Thanks :)

I especially like the last one

Tobias

yeah, that one is easy to describe and it has the same property I mentioned with intersecting open sets

Orange Harvester

3:02 PM

@Tobias I did not understand your solution.

Frank Science

@GregRos The first one is with intermediate value property but nowhere bounded. In fact, in each interval, the image is $\mathbb R$!!

Tobias

@OrangeHarvester It was just an idea that might give an example

Orange Harvester

@Tobias Okay.

Tobias

the non-continuous solution to that functional equation are very strange (and only exist due to AC)

in fact, the non-linear ones

Orange Harvester

Aha. Interesting.

Conway's base-13 is ingenious.

Greg Ros

3:04 PM

Can you give an example? @Tobias

Tobias

@GregRos as they require AC, no :)

Frank Science

@GregRos Suppose $\{\,e_1,e_2,\dotsc\,\}$ is a Hamel basis of $\mathbb R$, and consider $f\colon\,(r_1,r_2,\dotsc)\mapsto r_1$.

Tobias

As far as I recall, they are related to ultra-filters on $\mathbb{R}$

Frank Science

At first, I doubted a function has intermediate value property if and only if it is the derivative of some differentiable function, but Canway's base-13 function just broke my illusion.

Eric Gregor

3:40 PM

Suppose one wants to prove that a finite $d$-dimensional vector space over $k$ has a unique topology. Suppose $e_1,\dots,e_n$ is a basis for $k$. Why on Earth would it suffice to show that any map $(\lambda_1,\dots,\lambda_n)\to \lambda_1 e_1+\cdots+\lambda_n e_n$, from $k^n\to V$, is a homeomorphism?

user58512

@EricGregor, homeomorphic bijection, no?

Eric Gregor

is the point that $V$ is homeomorphic to $k$, and there is a unique topology on $k$?

user58512

uh

did you get what I said about bijection

Eric Gregor

yes, it's a bijection

user58512

ok great

Tobias

3:43 PM

@user58512 part of being a homeomorphism is being bijective

Eric Gregor

the bijection is immediate anyway

Tobias

@EricGregor anyway, once you have a homeomorphism to a specific topological space, then clearly your original topology was "the same" as the one of that space

Eric Gregor

yes, i see now. i think i was being retarded. writing out the question makes me see that now...

Tobias

I don't see how you could prove that though

Eric Gregor

by induction on dimension. can you see the one-dim case?

Tobias

3:46 PM

@EricGregor I guess we need to fix a topology on the field to make sense of the question in the first place

Eric Gregor

we are assuming that $V$ is a topological vector space, so the vector space operations are continuous

that's a key assumption i should have noted

Tobias

right, but there is not a unique way to put a topology on the field, so we need to fix that

Eric Gregor

i think the point is, as i think you noted, that the topology on $k$ is fixed from the start

Tobias

right

Eric Gregor

and the point is that the topology is unique relative to the topology on $k$

Tobias

3:48 PM

then it is clear in the dim 1 case

Eric Gregor

then you do quotient kind of arguments

Rachel

4:07 PM

Is there an easier way of simplifying A+(B-(A-D))?

I feel like there should be, but my brain's on strike today >.<

user58512

A+B-A+D

Gustavo Bandeira

@Rachel wolframalpha.com/input/?i=Simplify+A%2B%28B-%28A-D%29%29

Tobias

seriously? WA in order to simplify that?

Gustavo Bandeira

@Tobias If she needs a procedure, she could check the step-by-step thing.

user58512

buy mathematica for $6000 and do it

Tobias

4:12 PM

If you can't simplify something like this, it means your brain needs a break

Rachel

@GustavoBandeira Thanks, I'm bookmarking that site for later :)

Tobias

go outside for a breath of fresh air before doing further work

Rachel

@Tobias My equation isn't quite that simple... it actually looks more like SUM(F56:F72)+(F24-(SUM(F56:F72)-SUM(E56:E72))) and my brain is tired :)

Tobias

as I said, if your brain is tired, take a short break before doing more

Gustavo Bandeira

@Rachel I would make a little joke. XD

Tobias

4:14 PM

a breath of fresh air and drink some cold water

Rachel

@Tobias Then I'll forget where I'm at >.< I'm trying to automate an Excel report, and only have the final report data to go by. I don't have access to the actual person who knows the formulas for this report.

Gustavo Bandeira

Or...

Rachel

But I probably do need a break right now :)

Gustavo Bandeira

A nice use for venn diagramms

Tobias

Man, I can't wait for my Aeropress to arrive so I can make some proper coffee again. The machine here is just impossible to use for less than 10 cups, and I don't feel like paying for everyone else's coffee

Orange Harvester

4:25 PM

@GustavoBandeira I like cold coffee (and on occasions when I am feeling generous with whipped cream)

user58512

cold coffee?

I have never heard of such a thing

Orange Harvester

@user58512 iced coffee?

@Rachel I am copying your profile page about laziness etc. :P

Gustavo Bandeira

►

Someone downoted my question.

0

Q: What are the theorems on the inevitability of some kind of order in large sets?

I've read Paulos' A Mathematician Plays the Stock Market: The problem is that if you look hard enough, you will always find some seemingly effective rule that resulted in large gains over a certain time span or within a certain sector. (In fact, inspired by the british econom...

But it's really not so dumb.

user58512

it's a good question

Gustavo Bandeira

I know that Ramsey was one of the guys who developed it.

But the text say mathematicians over the last half century have proved a variety of theorems on the inevitability of some kind of order in large sets.

user58512

4:32 PM

Szemerédi regularity lemma

In mathematics, the Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost randomly. introduced a weaker version of this lemma, restricted to bipartite graphs, in order to prove Szemerédi's theorem, and in he proved the full lemma. Extensions of the regularity method to hypergraphs were obtained by Rödl and his collaborators and Gowers. Formal statement of the regularity lemma The formal statement of Szemerédi's regularity lemma requires some definitions. Let G be a g...

Gustavo Bandeira

So, I guess that other mathematicians also worked on something like that.

@user58512 You should post it as an answer.

Rachel

4:45 PM

@OrangeHarvester lol sure, you're welcome to it. I wrote that originally in a response to a Programmers.SE question (now deleted) that was asking how that quote from Larry Wall applied to programmers, and liked it so kept it around :)

Orange Harvester

@Rachel Cool. :-) Copying/Stealing should also be included in that list. :-)

Rachel

All I ask is if someone gets famous or makes money of something I said, be sure to give me proper credit ;)

Orange Harvester

will.do.so

Chris's sister

5:05 PM

Hi

Hi

Hi

Hi

I wonder how to easily prove that $\lim_{s\to 0} \int_0^1 (\Gamma (x))^s\space\mathrm{dx}=1$

user58512

that looks really tough

oh wait, can't you just take s=0?

or does the integral diverge for all s>0?

I think the integral converges for all 0 <= s < 1, if you can show that you can just use s=0.. but that may be difficult....

Chris's sister

5:15 PM

@user58512: $\int_0^1 (\Gamma (x)) \space\mathrm{dx}$ diverges ...

user58512

I keep typing s=1 instead of s=0 by accident

Chris's sister

@user58512: hehe

Probably it wouldn't be bad to split the interval and then to use the uniform convergence.

user58512

6:35 PM

hi

Tobias

hi

user58512

I'm bored any thoughts?

Tobias

do some math

user58512

do you know about irrational numbers?

Tobias

know what about them?

user58512

6:50 PM

I want an overview of what's known about it

Tobias

I don't think there is one unless you can be more specific

user19161

Ah, so many Rachels on SE. So far I have chatted with 3 of them.

user19161

@Rachel A+(B-(A-D))=A+(B-A+D)=A+B-A+D=A-A+B+D=0+B+D=B+D

user19161

Welcome back @jonas bro!

Jonas Teuwen

Hi.

@robjohn Good morning.

@JasonBourne Yes @WillJagy talked to me yesterday, I totally disagreed and then I slept and then a bit OK.

robjohn

6:56 PM

@JonasTeuwen Hey there... later morning now, but still morning.

Jonas Teuwen

It is like 8PM.

robjohn

@JonasTeuwen It's even later morning there :-)

Jonas Teuwen

I will start working on paintings now.

@robjohn Or very early...

robjohn

@JonasTeuwen fingerpainting?

Jonas Teuwen

No, inverse scattering 8-))).

user19161

6:57 PM

Deep stuff bro.

Jonas Teuwen

Got contacted by a professor. Exciting stuff.

robjohn

@JonasTeuwen perform an inversion map on the paint outside the can to get it back into the can

Jonas Teuwen

@robjohn 'I put your Rembrandt in the can!'. Was not so hard.

I think they'd kill me.

robjohn

@JonasTeuwen ouch...

Jonas Teuwen

8-).

user58512

6:59 PM

@Tobias, what are the important methods and what theorems are known

Tobias

@user58512 still, way too broad to have an answer

Jonas Teuwen

Must be a philosopher!

user19161

Today I answered 4 lhf and so far got 0,1,2,3 votes for them.

Ethan

What is the complex conjuguate of a dirichlet character?

user58512

@Tobias, why

@Ethan, it's a group homomorphism so just negate the input

Tobias

7:01 PM

@user58512 because the subject matter is too large

user58512

@Tobias, how can i get some idea about it then?

user19161

@Tobias If it is too large, we chop it up into disjoint union of intervals.

Tobias

@user58512 by figuring out what particular type of things you are actually interested in

Ethan

Can you say that slightly differently? I don't understand your terminology

user58512

@Ethan, why do you care about dirichlet characters?

user19161

7:03 PM

@user58512 That boy is the next Ramanujan dude!

robjohn

@JonasTeuwen are you insulting people again? :-D

Jonas Teuwen

@robjohn Yes!

user19161

@robjohn Are you being sarcastic again?

user19161

I just got two more "ass" flags.

Jonas Teuwen

Little stars?

user19161

7:06 PM

So guys, remember. Let today be the NO USING ASS DAY.

user19161

Come on, people flag every little thing these days.

Greg Ros

It's interesting how so very often in math people invent concepts to serve their intuition, and then realize that these concepts bring them to unintuitive conclusions.

user19161

Yes, that is because intuition is incomplete and imperfect, like life itself.

user58512

@Ethan, are you doing primes in arithmetic progressions?

Jonas Teuwen

Yes, so it is a good definition!

robjohn

7:09 PM

@JasonBourne was that a cutting jest?

Jonas Teuwen

@robjohn I received a cute book today.

It is 'Distributions and operators' by Grubb.

I think you might like it too.

robjohn

@JonasTeuwen is it available on amazon?

Ethan

@user58512 Given some character modulo a if I sum that character over the values coprime to a less then a can that sum be given in terms of another number theoretic function

Jonas Teuwen

@robjohn I think it is, but I got it for 25$ as it was available on SpringerLink :-))).

user58512

I don't know

robjohn

7:11 PM

@JonasTeuwen I was just looking at it on SpringerLink :-)

Jonas Teuwen

@robjohn Good.

I first found it while I was in a retirement address.

The dude was talking about ordinal numbered dimensions or what the hell.

robjohn

@JonasTeuwen It looks nice. I see it deals with some $\psi$DO calculus...

but $70...

Jonas Teuwen

Very right dear Sir (that is why I linked it).

Oh?

robjohn

eBook for $55

Jonas Teuwen

@robjohn Does the university you work at have access to SpringerLink (UCLA? They should).

robjohn

7:13 PM

@JonasTeuwen That is how I am looking at SL

Greg Ros

@JasonBourne I think what I take from it is that just because you have something in mind when you build something, doesn't mean what you had in mind and what you get are the same thing.

Jonas Teuwen

@robjohn Then you should be able to get it for 25$...

(printing on demand)

robjohn

@JonasTeuwen how is that different from an eBook?

Jonas Teuwen

@robjohn I just... discovered that it is a woman.

@robjohn That it is printed and bound in a paperback?

But the ebook is also free when you have access to SL? So why buy.

You Sir, are making no sense.

"And it has no regularity."

That sentence is so full of doom doom.

robjohn

@JonasTeuwen I see the "MyCopy Softcover Edition" for $25, but I don't see anything for free.

Jonas Teuwen

7:25 PM

@robjohn I've got that one!

That is to me only available when I can download the chapters in .pdf myself :-).

robjohn

@JonasTeuwen The "myCopy" edition?

Jonas Teuwen

If you have never bought such a thing, I can make you a picture, it is quite decent quality.

Yes.

You get a paperback with colored cover (like the standard one) and laser printed pages.

All quite OK. I think it is a good deal for 25$ as it is a real book (and printing is not free either).

robjohn

Ah, I see, the chapters are available individually in PDF online

Orange Harvester

Programmer in a meltdown

@robjohn That is how I would get stuff from my university. I would download the individual chapters and then string them together using ghostscript or something.

robjohn

@OrangeHarvester Yeah, it just took me a while to find the right page and realize that all the chapters were there.

Orange Harvester

7:30 PM

@robjohn Ahh, nice.

Rachel

@JasonBourne Thanks, I knew it was something simple like that but I had been looking at numbers for far too long and my brain was on strike. It didn't help that I wasn't actually working in easy units like "A" or "B" :)

Orange Harvester

@JasonBourne I am back to Chromium for now. Mainly for memory footprint.

Firefox must have some incredibly bad memory leaks.

Theorem

Hi everyone 1

Orange Harvester

@Theorem Hi

Theorem

@OrangeHarvester wassup ?

Jonas Teuwen

7:36 PM

@robjohn But 25$ seems like they try to 'be a better publisher' with all the crap going on 8-)).

Orange Harvester

agreed.

@Theorem Nothing, want to do some problems on rings but can't.

Theorem

@OrangeHarvester : hmm , sounds great .

Orange Harvester

@Theorem not really, not when you suck as much as I do. What are you doing?

Joey Morani

Hey. Could anyone explain how I would find the gradient of a straight line with the equation 25x + 5y = 3 without telling me the answer? :) I worked it out to be -5. Is that correct?

Orange Harvester

@JoeyMorani What is the definition of a gradient?

Theorem

7:39 PM

@JoeyMorani : u are right .

Joey Morani

Gradient of a line on a graph.

Thanks! :)

Theorem

@OrangeHarvester : i am going to start preparing for my quantum physics/ electrodynamics exam

Orange Harvester

@Theorem heh, nice. Quantum physics + electrodynamics in the same course?

or is it qed?

Theorem

@OrangeHarvester : its not qed , but its sort of a packed course .

Orange Harvester

@Theorem i see.

Greg Ros

7:47 PM

Why don't they just measure angles in circles.

E.g. 1/2 of a circle, 1/3 of a circle...

Gustavo Bandeira

Pearls of reasoning: youtu.be/YGcLdzQeEck

Orange Harvester

@GregRos May be because of arc length parametrization.

robjohn

@JonasTeuwen I got the PDF (463 pages in 3.7 MB)

Gustavo Bandeira

@robjohn You're in piracy?

user19161

@Rachel We spoke before in the Eng room. I remember the Eng question you posted, about you and your boyfriend. Are you guys getting married soon?

Jonas Teuwen

7:53 PM

@robjohn Got it too.

@GustavoBandeira No, you can just download it.

robjohn

@GustavoBandeira No. got it via SpringerLink

@JonasTeuwen I think there were some blank pages not in the PDFs... page viii for example

user19161

@GustavoBandeira I do view books in PDF before deciding whether to buy them or not.

user19161

It's not enough to check amazon and google books.

Orange Harvester

@robjohn Generally people remove blank pages from PDF to prevent the doubts of corruption. Else, blank pages in PDF have a notice, "intentionally left blank".

Jonas Teuwen

@robjohn There is no page viii 8-).

(in my book)

user19161

7:54 PM

@OrangeHarvester That is so silly.

robjohn

@JonasTeuwen even in the print version?

Orange Harvester

@JasonBourne It is not!

Jonas Teuwen

Yes.