Mathematics - 2013-03-30 (page 2 of 4)

@user67848 I think sometimes it's hard to come up with a decent title, i.e., one which is descriptive of the content to come and isn't the question itself.

Dominic Michaelis

12:40 PM

The user said oh silly mistake but doesn't fix it :(

Git Gud

@DominicMichaelis I don't think the FAQ or the site itself transmits the idea that MSE is also supposed to be a repository for questions and answers. Given that, what would be the point once the OP is clarified?

Dominic Michaelis

cause half of the answers says the limit is pi and the other half says it is infty (or doesn'T exist)

I have to go working now i am afraid, see you later

Git Gud

@DominicMichaelis See you. Just to be clear: I agree with you.

Dominic Michaelis

See you too :)

pourjour

1:28 PM

hi

Ananth

2:42 PM

hi

user19161

I am now the black square.

user19161

@PeterTamaroff I like Rudin's proof best. It is also stated in a nice form.

Ananth

That's the name for a bug on Mac

black square

amWhy

Hello all! Good morning (my time)...

Oh, I didn't "recognize you" Jasper ;-) "Back to the black"

Carpediem

2:59 PM

Can someone help me with this ? math.stackexchange.com/questions/346519/…

amWhy

3:09 PM

This question seems to be right up your ally, @Jasper

Carpediem

3:22 PM

math.stackexchange.com/questions/346519/…

kram1032

3:45 PM

Hi

Quick question:

Is there any "middle ground" between topology of manifolds and network topology?

Like, topology that allows some structure like two euclidean planes, connected by a single edge

or a network that has continuous pieces between edges instead of nodes. - Like actual topological spheres

Or a Möbius strip that is touched by a sphere in exactly one point. Stuff like that

Carpediem

4:06 PM

Can someone help me with this? math.stackexchange.com/questions/346545/…

@kram1032

kram1032

4:17 PM

Sorry, Carpediem

Wait, that is a different question than before

Jorge Fernández

hi fellas

kram1032

hi

still, sorry @Carpediem I'm not sure

@Jorge do you know of an extension / a branch of topology that can deal with connections of manifolds with networks?

(Examples above)

Jorge Fernández

sorry, i'm i high school

kram1032

heh

Jorge Fernández

4:37 PM

what are some basic examples of non-constructive existance proofs?

kram1032

The Intermediate Value Theorem or the Mean Value Theorem

or well

They can used to proof things without construction

Michael Corleone

You can prove with the axiom of choice (so non-constructive) that if $E\subset\mathbb R$, $E$ is bounded and $E\neq \emptyset$, then there is a sequence completely in $E$ which converges to $\sup(E)$

kram1032

non-constructive simply means, you don't find an explicit example but you proof that it has to exist

Jorge Fernández

is the proof of ramsey's theorem constructive?

Alex J Best

I think there is more than one?

Charlie

4:45 PM

Hello

kram1032

It's rare that there is just a single proof for things, right?

Jorge Fernández

@AlexJBest you are studying discrete mathematics?

Hi Charlie

Hi

@JorgeFernández Yes

4:46 PM

Is that the name of the program?

kram1032

@Jorge, Michael's example was pretty good.

Alex J Best

@JorgeFernández Yeah, www2.warwick.ac.uk/fac/sci/dcs/admissions/undergraduate/dmaths

kram1032

He stated, that there's always a sequence of numbers that lies in a finite interval and converges against the supremum of that interval

Charlie

@MichaelCorleone hi :)

kram1032

but he didn't construct any such sequence. He simply used the Axiom of Choice

Michael Corleone

4:49 PM

@Charlie hi, how are you?

Jorge Fernández

what is undergrad called in england? @AlexJBest

Alex J Best

@JorgeFernández Not sure what you mean, we call undergraduates undergraduates :)

Jorge Fernández

and these programs last 3 years?

kram1032

@Alex you don't happen to know of a field that can deal with intersections of discrete and continuous systems in a meaningful way?

I feel like that's related to my topology question above

Charlie

@MichaelCorleone I'm fine, and tou?

Alex J Best

4:51 PM

@JorgeFernández Generally a BSc does yeah, though it's reasonably common to do a masters in maths in 4 years here too.

Michael Corleone

@Charlie me too

Jorge Fernández

so you can do the bachelor and master in only 4 years?

Charlie

@MichaelCorleone great! Lot of work?

Alex J Best

@kram1032 I guess it depends on the system, what sort of thing did you have in mind?

Michael Corleone

@Charlie Yes, but I don't really mind. I'm enjoying it. And you?

Alex J Best

4:53 PM

@JorgeFernández Yeah it's pretty similar to doing a BSc they just have an extra year where they do more#.

Jorge Fernández

nice, are you going to do that?

kram1032

@Alex stuff like connections between continuous and discrete topologies. E.g., say, a topological sphere which has one point that is shared with a network. For the sphere it would be a point, for the network, it would be a node

Charlie

@MichaelCorleone me too :)

kram1032

I hope that makes sense

Michael Corleone

@kram1032 Yes, the idea is to consider for each $n\in\mathbb N$ a set $E_n=\{x\in E|\sup(E)-\frac{1}{n}<x\leq \sup(E)\}$. These sets are all not empty, otherwise $\sup(E)$ wouldn't be an upper bound, so by the axiom of (countable) choice we can pick an $a_n\in E_n$ for each $n\in\mathbb N$. This gives us a sequence which, by the squeeze theorem, converges to $\sup(E)$.

kram1032

4:57 PM

@Jorge does what @Michael just wrote help you?

(it's about your question concerning non-constructive proofs)

Michael Corleone

Probably not :P. He seems to be more interested in discrete maths

Jorge Fernández

yes, it helps

Alex J Best

@kram1032 I don't know the name of such an area, but it seems likely it exists!

Jorge Fernández

i just wanted to see the difference

thank you

Michael Corleone

Ok good no problem

Alex J Best

4:59 PM

@JorgeFernández I'm not sure, I'd like to do a masters, not really sure where yet though!

Jorge Fernández

Is the difference that in a constructive proof you know it exists and exactly how to find it?

kram1032

@Jorge in a constructive proof, you proof by showing an example

in a non-constructive one, you proof by saying it's within the rules of the system so it has to exist

(eventhough you don't necessarily know how it would look like)

Cortizol

just wondering, here math.stackexchange.com/questions/327110/… in title we have: using integration by parts, but @user43418 bounty that question and said: "How can we show the above property using sinAsinB and using Euler's formula?", huh

user19161

@Charlie I am black too, like you.

3

kram1032

@Alex Thanks. It's always so hard to find stuff about a topic if you don't know the name.

@Alex I essentially had some system in mind where you connect edges of a network with smooth topologies rather than nodes and then use it as a sort of neural topology instead of network. Or maybe a Markov process that happens on such a space. Stuff like that

Alex J Best

5:03 PM

@kram1032 math-atlas.org/welcome.html is normally an okay place to try for finding things.

Charlie

@JasperLoy :DDD

kram1032

My quick and dirty research brought me to "General Topology" and then, unrelatedly, to "pointless topology" (which is a hilarious name) - Wikipedia is such a distracting place with all its internal or external links

@Alex oh, that looks like a nice, if slightly ugly place. Thank you :)

Alex J Best

@kram1032 Haha yeah, it's another place to easily get lost reading things, the website is a bit old though!

Charlie

@kram1032 yep, indeed :)

Alex J Best

I end up doing this a lot xkcd.com/609

kram1032

5:07 PM

@AlexJBest this a million times

Michael Corleone

Haha I've ended up before with 7 firefox windows open with each more than 100 tabs

kram1032

haha, I usually go with one window with more than 700 tabs

I've been hopping back and forth between different ideas. I wish I had the actual deep mathematical understanding to try to explore them myself.

For instance, I found a paper on "Lineal", a linear-algebraic lambda-calculus, which essentially combines lambda-calculus with the notions of linear algebra, and lambda terms are basis vectors

From there I thought "what if this works with process calculi too, like the pi-calculus"

And my next thought was that, given such a system, you could probably define basis-transforms that transform between the lambda- and pi-calculus

user20997

what is meant by "complex-linear group action"? I have a linear group G acting on the space $\mathbb{C}^6$.

kram1032

action in that sense. I stumbled over that just recently. What was that now again

user20997

does it meant that g.(x+y)=g.x+g.y for g in G and x,y in C^6?

kram1032

5:15 PM

group action is a description of symmetries of a group

Carpediem

Hello.

kram1032

it at least sounds like that, @user20997

Carpediem

Can someone help me with this ?

2

Q: Show $N \sin w_{0} t+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin w_{1}t=N(1-e^{\frac{-t}{T}})\sin w_0 t+ \mathrm{O}(1)$

We have $$x(t)=N \sin w_{0} t+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin w_{1}t$$ and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$. I need to show that $x(t)=N(1-e^{\frac{-t}{T}})\sin w_0 t+ \mathrm{O}(1)$. I obtained $x(t)=N\sin w_0 t+\frac{N^2}{\delta_1+N^2}e^{-\frac{t}{T}}\sin w_1...

user20997

I know what group action is, I just don't understand what is meant by "complex-linear"

kram1032

@user20997 linear certainly has to do with forms like a(x+y)=ax+ay. Complex here probably means that a can be out of the Complex numbers

@user20997 so yeah, your idea up there seems right

user20997

5:19 PM

ok, just could not find any references in the literature

kram1032

@user20997 does this help? en.wikipedia.org/wiki/Linear_complex_structure

κρανίοπεριπολία

6:06 PM

6 hours ago, by Jasper Loy

@κρανίοπεριπολία In French texts translated into English, 0 is both positive and negative. Just a linguistic convention.

@JasperLoy could you elaborate a little more on this liguistic convention please :)

user19161

@κρανίοπεριπολία Well, it's just that I have a few French translated into English books that define "positive" as greater than or equals 0 and "negative" as lesser than or equals 0, that's all.

user19161

But in originally English texts, this is never the case.

user19161

I say this from the 9000 books I have seen.

κρανίοπεριπολία

It doesn't really make logical sense either for 0 to be both?

user19161

@κρανίοπεριπολία So if we use such a definition, 0 is both positive and negative. Just a matter of different definitions.

user19161

6:14 PM

The editors page says I have done 500 edits. Does it mean the Copy Editor Badge is on its way? I have a feeling the number on that page is not the correct one.

κρανίοπεριπολία

The check is always "in the mail." ;-)

Carpediem

Does anybody know how to solve this?

0

Q: Approximation and boundness

I have a question regarding a certain approximation. We have $$x(t)=N \sin w_{0} t+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin w_{1}t$$ and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$. I need to show that $x(t)=N(1-e^{\frac{-t}{T}})\sin w_0 t+ \mathrm{O}(1)$. I obtained $x(t)=N\sin ...

approximation bounded-variation

κρανίοπεριπολία

Here's one where the word "usage" is misspelled ;-)

6

Q: Are there rules in the useage of prepositions in Math?

It is often to use prepositions in various expressions. E.g. $2$ is in the set of natural numbers $\mathbb N$ The symmetric group on 3 letters $S_3$ is the group consisting of all possible orderings of the three letters ABC i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 eleme...

soft-question terminology article-writing

yet it has 6 upvotes and 4 answers.

@GN

Carpediem

6:36 PM

@κρανίοπεριπολία Can you help me with my problem?

κρανίοπεριπολία

I would, if I could.

@Carpediem sorry.

anon

6:52 PM

@user20997 Complex-linear means that for any g in G, we have g(ax+by)=a(gx)+b(gy) for all vectors x,y and complex scalars a,b, which essentially means that it's an additive map and commutes with complex scalar multiplication. any complex vector space can be regarded as a real vector space, and then the group of real-linear maps is even bigger than the one of complex-linear ones.

Cortizol

where can i find algebraic proof that heptadecagon is a constructible polygon? (I know its Fermat number, but without that)

anon

how many sides is that again?

Cortizol

17

anon

@Cortizol uwgb.edu/dutchs/pseudosc/17-gon.HTM (pretty sure I saw a slick proof of Gauss' using cyclotomic fields, I will try and find it)

Cortizol

@anon i mean something like en.wikipedia.org/wiki/…, thanks for helping me :)

anon

6:59 PM

hmm, this looks pretty cool. you can find other resources by googling combinations of "17-gon," "gauss," "cyclotomic," "constructible," "galois," etc.

Cortizol

@anon ok, i will try :) thanks for this, i will check it !

Cтарый Джон

@anon Wasn't that the discovery that ultimately lead to Gauss choosing maths as a career-path?

anon

I may have read that, don't remember

Cтарый Джон

Hmm W|P seems to say it made him choose maths instead of philology

user19161

@CтарыйДжон Never seen that abbreviation.

anon

7:11 PM

@CтарыйДжон I'm looking at this and I get ${\rm Tr}(A)=(p-1)\sum_{k\ge0}(-1)^kf_k$; do you see what I may be doing wrong? I figure ${\rm Gal}({\bf Q}(\zeta_p)/{\bf Q})=({\bf Z}/p{\bf Z})^\times$

Dominic Michaelis

i first thought of wolfram alpha

@jasper why are you sad ?

Cтарый Джон

@JasperLoy I made it up :)

user19161

@DominicMichaelis Where did I say that?

Dominic Michaelis

black square, black = sad

Cтарый Джон

@anon will take a look ...

user19161

7:13 PM

@DominicMichaelis No, the colour just means whatever I like to look at at the moment.

user19161

However, I think I also have reason to be upset, but that is a secret!

Dominic Michaelis

i hope it's a sexy women in a short black dress ;)

user19161

Because yesterday, someone downvoted 4 posts of mine in 2 min.

Dominic Michaelis

oh that bitch

κρανίοπεριπολία

But it was reversed, right?

anon

7:14 PM

So ${\rm Tr}(\zeta_p^l)=\sum_{\sigma=1}^{p-1}\zeta_p^{\sigma l}$ is $p-1$ if $p\mid l$ and $0$ otherwise.

user19161

Yes, it was reversed. Does not matter who it is, I can't be sure either.

Cтарый Джон

@anon I would agree with the Galois group

@anon I think so too

user19161

Even if not reversed, also does not matter. There are more important things in life, such as health.

κρανίοπεριπολία

You can't please all of the people all of the time.

user19161

Anyway, I think Gauss is so awesome.

Ilya

7:18 PM

@CтарыйДжон hey, nice to see that someone uses a proper alphabet :)

user19161

Anyway, instead of using more complicated LaTeX, I just write my multiline equations here using enter for the next line.

κρανίοπεριπολία

@Cortizol Didn't he want a heptadecagon engraved on his tombstone?

Dominic Michaelis

@jasper that's cheating

Cтарый Джон

@Ilya II changed it just for an experiment - now I am stuck with it for a month :)

Ilya

:D

user19161

7:19 PM

@DominicMichaelis Well, we are not writing a LaTeX document when we type an answer here!

Ilya

happened to me once with "il y a"

user19161

For one, we don't start with \documentclass{article}, lol.

Cтарый Джон

@Ilya nice :)

Dominic Michaelis

article is a no go anyway you should use scrartcl btw :D

user19161

@Ilya I remember you got upset when I wrote ilya and not Ilya, but that is how I ping everyone.

user19161

7:22 PM

@CтарыйДжон I might change mine again soon.

Cortizol

@κρανίοπεριπολία I heard that :)

Cтарый Джон

@JasperLoy to what?

user19161

@CтарыйДжон Hmm still thinking.

anon

@CтарыйДжон robjohn would love to help you change it sooner if you desire :)

user19161

@anon Yes, I think mods can do that under "special circumstances"!

user19161

7:25 PM

In this case, it would be that the name is too hard to type!

Cтарый Джон

@anon Yes - I might ask him, as people get a bit annoyed with difficulties pinging my present name

anon

last april 1st I changed my name and gravatar to match robjohn's

Cтарый Джон

(not that many people want to ping me!)

user19161

@CтарыйДжон Yes, they might get so annoyed and start to downvote you!

user19161

@anon Yes, and it had to be reversed!

user19161

7:26 PM

@DominicMichaelis Do you talk to Charlie on FB?

Cтарый Джон

Hmm April 1st is coming around again soon ... wonder if people have any fun pranks planned?

user19161

I once knew a beautiful girl called April.

user19161

We were not in love, but we went out together very often.

Cтарый Джон

@JasperLoy possible puns along the lines of "was April your 1st" ... but probably bad taste!

κρανίοπεριπολία

or last :-D

user19161

7:31 PM

I went out with her boyfriend very often too.

user19161

They are now married with two kids.

user19161

My best friend is also married with two kids.

user19161

@κρανίοπεριπολία Oh you should consider changing your name too to something more easily pingable.

κρανίοπεριπολία

13 mins ago, by Cтарый Джон

(not that many people want to ping me!)

@JasperLoy same answer ;)

user19161

@κρανίοπεριπολία Haha, I think "ping" also has a sexual meaning...

κρανίοπεριπολία

7:41 PM

ping-pong?

user19161

I am not too sure actually, will need to look it up!

user19161

But pong pong is a fruit, a poisonous one.

user19161

And ping pong is table tennis.

κρανίοπεριπολία

ping n. short sharp ringing sound.

-v.i. make this sound.

Charlie

Hi

κρανίοπεριπολία

7:44 PM

hi

Jayesh Badwaik

@Charlie ....

user19161

@JayeshBadwaik What do the four dots mean? Some secret?

Charlie

@κρανίοπεριπολία hi skull @jayesh I lost connection

Jayesh Badwaik

@JasperLoy yeah.....

user19161

@JayeshBadwaik OK, so secretive!!!

Jayesh Badwaik

7:46 PM

@Charlie I see. What about now?

Charlie

@jasper hi

Jayesh Badwaik

@JasperLoy :P

user19161

@Charlie Hey hey.

Jayesh Badwaik

@JasperLoy Wassup?

user19161

I imagine that the graph of secrets shared between MSE users looks very complicated...

Charlie

7:47 PM

@JayeshBadwaik it's fine now, jayesh badwaik

user19161

Each person has many secrets, and shares only some secrets with some people.

κρανίοπεριπολία

Define "secret."

Cortizol

Some hint for $$\int\limits_0^1{\int\limits_0^1{\frac{{\ln\left({2-xy}\right)}}{{1-xy}}dxdy}}=\frac{5}{8}\zeta\left( 3\right)$$? :) this looks to famous to me. Maybe I saw it somewhere

Jayesh Badwaik

@JasperLoy Yes, also, there might not be circuits, for example, two person may know the secret about the third person, but they do not tell it to each other. So,

user19161

@κρανίοπεριπολία Well, you can define it anyway you want.

user19161

7:49 PM

@JayeshBadwaik Yes, how interesting!

κρανίοπεριπολία

@JasperLoy can you keep that defintion a secret?

Carpediem

@JasperLoy Can you help me with this? math.stackexchange.com/questions/346726/…

user19161

@κρανίοπεριπολία I have not defined it in anyway.

user19161