Mathematics - 2012-08-01 (page 3 of 4)

I mean there are always people that seem to think they are better at it then me but don't even know Maxwell's laws. That is quite peculiar, but sure. I believe them. After all I am just a retard.

Peter Tamaroff

I have new glasses with "Reflex" and when I look through them in a very acute angle, objects appear with a red-blue (like the $3D$) halo. Blue appears always in the right and red in the left

Jonas Teuwen

That's like too applied. I have only seen a lens in the section museum.

But, I think I know what it is.

Is it polycarbonate or glass?

Peter Tamaroff

@JonasTeuwen Poly

Glass is not longer used AFAIK

Jonas Teuwen

Okay, there are several coatings on it, to remove glare and to make it more resistant to scratches.

@PeterTamaroff In some countries it is and you can probably still order it.

Peter Tamaroff

@JonasTeuwen Yes.

@JonasTeuwen It mean, it is dangerous!

Jonas Teuwen

8:04 PM

Yes, but you can make the glass way harder, such that the blow actually would already break your skull 8-).

@PeterTamaroff I think this can explain it: yeah this.

Peter Tamaroff

@JonasTeuwen If you see the reflection of light in the outer side of the poly, it is greenish. That is the reflex.

Jonas Teuwen

@PeterTamaroff You have the following thing: the refractive index which denotes the diffraction of a beam of light.

And the wavelength of the light.

Normal light contains many different wavelengths, and if you throw it on some crystal you get a rainbow right?

Peter Tamaroff

@JonasTeuwen Yes.

Jonas Teuwen

If you make the layers very thing, it can be even more spectacular when they are close to the wavelength of light, and if you add several.

So you are probably in the right conditions to see that particular type of light.

If you want to analyse it it would be quite hard, the glass is not completely flat, the layers are probably quite proprietary and so on.

Peter Tamaroff

@JonasTeuwen I'm a wannabe cyborg!

Jonas Teuwen

8:07 PM

@PeterTamaroff KICKASS.

I hope it made some sense.

Peter Tamaroff

@JonasTeuwen set-filter: RGB=101

Jonas Teuwen

I have only done experiments in the undergrad.

After that. I denounced experiments.

But I got $\geqslant 95\%$ on the lab exercises.

Peter Tamaroff

@JonasTeuwen What applications does quantum optics have?

Jonas Teuwen

How would I know? I just compute.

Usually when the light is of very low intensity.

Peter Tamaroff

@JonasTeuwen Did you ever "accidentaly" burn an annoying classmate? I would lose a $5\%$ on that

Jonas Teuwen

8:10 PM

Nope.

Only 5%? We would get kicked out.

Peter Tamaroff

@JonasTeuwen Hahah JK

Jonas Teuwen

"Sorry for shining this nice 10W laser into your eyeball and that I burned a hole through your head. I hope you don't mind too much."

Peter Tamaroff

@JonasTeuwen See this

Matt N.

@anon ayt? I need help with inverse limit...

Jonas Teuwen

@PeterTamaroff So... applied...

anon

8:13 PM

@MattN. what's up

robjohn

@MattN. Limit of an inverse function or limit of a reciprocal?

Matt N.

@anon In Dylan's answer here I'm trying to understand what's going on. Let me post what I have so far.

@robjohn $\varprojlim_n A/A_n $, the inverse limit of an inverse system.

Peter Tamaroff

@MattN. Everything is upside down

Matt N.

@anon I'm not sure $p^k B$ is really zero. (in (ii))

anon

it isn't

Matt N.

8:16 PM

Yeah, that's what I suspected.

But $$ p^kB = \{0\} \oplus \cdots \oplus \{0\} \oplus p^k\mathbb Z/p^{n}\mathbb Z \oplus p^{k}\mathbb Z/p^{n + 1}\mathbb Z \oplus \cdots$$

So the $n$-th component looks like $p^k m \mod p^{n}$.

anon

mmhmm

Matt N.

And the $n$-th component in $A$ looks like $m^\prime \mod p$.

What's wrong with this?

anon

there's a difference between $A$ and $\alpha(A)$.

Jonas Teuwen

The second one has an $\alpha$ extra!

Matt N.

Oh! Yes! I multiply the $n$-th component by $p^{n-1}$.

But what does the inverse map do? I don't have division.

Do I really do $\alpha(A) \cap p^k B$ instead of $A \cap \alpha^{-1}(p^k B)$?

anon

8:21 PM

yes, like I said in my comment. the latter doesn't even make sense does it?

because $\alpha:A\to B$

nvm, you edited it

Matt N.

I'm trying to see what $\alpha^{-1}(p^k B)$ looks like.

anon

what makes you think $p^kB$ is contained in the image of $\alpha$?

Matt N.

@anon That I don't know but the topology induced by $Y$ on $X$ given a map $f: X \to Y$ is open sets are $f^{-1}(O) \cap X$ where $O$ open in $Y$, no?

anon

wait, yes, that makes sense. I'm being foolish; inverse images work even for sets that seep out of the image

Jonas Teuwen

How are (rational) roots usually defined? As the inverse of ...?

Matt N.

8:26 PM

Yes.

Jonas Teuwen

(or real numbers)

Matt N.

@anon So: what does the inverse image of $p^k B$ look like? (because I don't understand why I should be allowed to twist the definition of induced topology to use $p^k B \cap \alpha (A)$ instead)

anon

the inverse image of $p^k B$ should be like$$\underbrace{0\oplus\cdots\oplus0}_k\oplus \Bbb Z/p\Bbb Z\oplus\Bbb Z/p\Bbb Z\oplus\cdots$$ right?

Matt N.

How did you get to that?

Peter Tamaroff

@anon What is this theory?

Jonas Teuwen

8:30 PM

@anon That looks scary, man... need to go to the potty now (I love my potty).

Old John

@PeterTamaroff p-adics done the abstract algebra way, I think

anon

@MattN: Fuckin magnets. @PeterTamaroff Somewhere in the intersection of topology, group theory, abstract / commutative algebra, maybe number theory

Old John

very different from the way I do them :)

Jonas Teuwen

@OldJohn With a nice glass of Scotch?

Old John

@JonasTeuwen Nah - wine tonight - run out of scotch :(

Peter Tamaroff

8:31 PM

@anon Quite a crossroad!

Jonas Teuwen

@OldJohn Wow, that's horrible, man. Hope you will be okay.

I have like 38 bottles left.

Not so much, but it is fine :-).

Old John

@JonasTeuwen Good grief! I have never owned that much - ever

Jonas Teuwen

Oh? Most people I know that know a bit about it have way more.

anon

@MattN. seriously though, it's a direct sum. you just take the inverse image of each component at a time

Matt N.

@anon But I don't understand what that is either because we don't have division.

How do you do it?

Peter Tamaroff

8:35 PM

@anon I think that terms such as "obvious" "just" "clearly" "at once" and similars have a psychological effect on the young mathematician.

anon

@MattN. $n$th component of $\alpha(A)$ equals $\alpha_n(\Bbb Zp / \Bbb Z)=p^{n-1}\Bbb Z/p^n\Bbb Z\subseteq p^k\Bbb Z/p^n\Bbb Z$ equals $n$th component of $p^k B$ when $n>k$

Matt N.

@anon Yes ok, so we have that the image of the $n$-th component of $A$ lies in the $n$-th component of $p^kB$. But I still don't see how to take the inverse image from there.

Henning Makholm

All my numerological ambitions fail :-/. I missed 32,768, and here I am at 34,587 rep. If I'd just checked it two upvotes earlier ...

anon

@MattN. If $f:X\to Y$ and $f(X)\subseteq U\subseteq Y$ then $X\supseteq f^{-1}(U)\supseteq=f^{-1}(f(X))\supseteq X$ hence $f^{-1}(U)=X$

Matt N.

@anon Oh me GAWD. facepalm Thank you so much!!

anon

8:45 PM

blegh. you get the point.

Matt N.

So the inverse image of $p^k B$ is just the whole space: $\mathbb Z / p \mathbb Z $ in the example.

anon

so long as $n>k$, yeah

Matt N.

No. You're right: $n-1 \geq k$.

Peter Tamaroff

@MichaelGreinecker Hey

Michael Greinecker

Hey

Peter Tamaroff

8:51 PM

@MichaelGreinecker Are you doing any maths at the moment?

Michael Greinecker

Not really, what's up?

Matt N.

All punk music songs sound the same : ) Reminds me of being still in high school.

Apparently, the lead singer died just yesterday.

^random fact of the day

Peter Tamaroff

@MichaelGreinecker Not much. I'm reading some Set Theory from Halmos' and yesterday I was looking for some Topolgy exercises. Dunno if you saw the $T_1$ topologies exercises proposed by @MarkDominus

@MattN. I'm listening to Jimmi now.

Michael Greinecker

@PeterTamaroff No. Where can one find them?

Peter Tamaroff

►

@MichaelGreinecker Sorry, I meant $\rm exercise \not{s}$

Matt N.

8:59 PM

@PeterTamaroff I'm listening to Salt n Pepa now.

Peter Tamaroff

@MichaelGreinecker It is as follows:

A $T_1$ space is a topological space $(X,\mathfrak I)$ in which everysingleton $\{x\}$ is closed. Show that for any set $X$ there is a unique weakest topology $\mathfrak I_1$ such that $(X,\mathfrak I_1)$ is a $T_1$ space.

Given two topologies $A$ and $B$, we say $A$ is weaker than $B$ if $A\subset B$

Michael Greinecker

@PeterTamaroff Got it.

@PeterTamaroff You want hints?

Peter Tamaroff

@MichaelGreinecker So basically we want to find a $T_1$ topology $A$ such that whenever $A'$ is another $T_1$ topology $A\subset A'$

This basically hinted me into thinking about intersections.

Michael Greinecker

@PeterTamaroff Here, the bottom up approach works better. What sets must be in the topology?

Peter Tamaroff

I know that the closure of the set $A$ can be either proven or defined as the intersection of all closed sets containing $A$. Then I think one can find this unique topology by thinking about the intersection of all $T_1$ topologies.

@MichaelGreinecker "Punctured spaces" so to say, that is $X\setminus \{x\}$ with $x\in X$.

Subsets of $X$ with finite complement?

Michael Greinecker

9:07 PM

@PeterTamaroff Yes, exactly. So what sets can you obtain from them by finite intersections and arbitrary unions?

Peter Tamaroff

@MichaelGreinecker Finite complements?

Michael Greinecker

@PeterTamaroff Why?

Peter Tamaroff

@MichaelGreinecker Say $A=X\setminus \{a\}$ and $B=X\setminus \{b\}$. Then $A\cap B=X\setminus \{a,b}\$?

Michael Greinecker

@PeterTamaroff Yes. And for unions?

Peter Tamaroff

@MichaelGreinecker A union might just "cover" the other one, so at most you get $X$.

Every $T_1$ topology is a subset of the cofinite topology.

Michael Greinecker

9:12 PM

@PeterTamaroff A superset.

Peter Tamaroff

@MichaelGreinecker Bah! I always get these inclusions backwards!

So the cofinite topology is the answer.

Michael Greinecker

@PeterTamaroff Yes. Congrats!

Jonas Teuwen

@OldJohn Glenn Gould. Playing!

Peter Tamaroff

@MichaelGreinecker But I have to prove that!

Old John

@JonasTeuwen Excellent!

Peter Tamaroff

9:14 PM

@MichaelGreinecker Could you reason that out?

anon

@PeterTamaroff you said yourself: whenever $A'$ is another T1 topology, $A\subseteq A'$ i.e. $A'$ is a supset

Michael Greinecker

@PeterTamaroff The basic facts to use are: Finite unions of finite sets are finite. a subset of a finite set is finite. The first one helps with finite intersections, the second one with arbitrary unions.

Old John

@JonasTeuwen What is he playing?

Peter Tamaroff

@MichaelGreinecker Arent arbitrary unions of finite sets countable?

Jonas Teuwen

@OldJohn Bach Goldberg Variations.

Peter Tamaroff

9:18 PM

Like $\{1,2,3,\dots,n\}$, $\{n+1,\dots,2n\}$ &c

Old John

@JonasTeuwen Lovely - I have a recording of him playing those

Michael Greinecker

@PeterTamaroff arbitrary unions of finite sets can be of any cardinality. But you want to understand unions of cofinite sets.

Matt N.

Bleh.

Peter Tamaroff

@MattN. What's up?

@MichaelGreinecker OK. So, recapping.

Matt N.

@PeterTamaroff Asaf just annoyed me. And I'm annoyed that I have any sort of reaction to what's going on on this site.

Old John

9:20 PM

@JonasTeuwen also a fascinating arrangement for string trio (purists might hate it)

Jonas Teuwen

@OldJohn I am not so knowledgeable about it yet :-).

Peter Tamaroff

@MattN. Asaf seems to be short tempered. It happened to me too.

Jonas Teuwen

@MattN. Asaf annoyed you? How! :-).

Peter Tamaroff

He didn't seem to react well to lots of questions.

Matt N.

@JonasTeuwen Well feel free to read his last comment.

Jonas Teuwen

9:21 PM

He never annoyed me, just made me laugh.

Matt N.

I can annoy him too though.

Just a sec.

Peter Tamaroff

": Could you please take it somewhere else? If you take it to Vienna somewhere late September, I'll be happy to see you discuss this at a bar. I'll even have several rounds of beer with the lot of you. "

Jonas Teuwen

That's quite a cool remark eh?

He asks you to not spam the comments to his post and come have a drink with him.

Peter Tamaroff

@JonasTeuwen That is prolly not his last comment.

XD

Matt N.

@JonasTeuwen What's cool about that? I think one should not barge into other people's "private" conversations.

Peter Tamaroff

9:23 PM

@MattN. Seems a friendly response to me.

@MattN. What are those "progress" bars?

Jonas Teuwen

@MattN. You might just be a bit stressed as I see nothing wrong with it.

@MattN. Well, if some people are having a loud conversation when I am in the train. In front of me. I ask them to stfu too!

Matt N.

@PeterTamaroff My exam prep. But I don't give a shit anymore. I've had it. Plenty.

Michael Greinecker

I'm in Vienna.

Peter Tamaroff

@MattN. I can get very fuzzy when I'm stuk with a problem. Maybe you're a bit mad at this inverse limit stuff?

@MichaelGreinecker I like trains. And what about the cofinite topology?

Can we try make a sketch of the proof?

Matt N.

@MichaelGreinecker Are you from there? I don't understand them when they speak! 0_o I guess if we really did go there we could all have a drink together I suppose...

Old John

9:24 PM

@MichaelGreinecker Hoping to visit Vienna from (a holiday in) Bratislava later this year

user19161

@JonasTeuwen Well, again, I have no equipment to do any sound recording for you now, I am technologically primitive!

Matt N.

@PeterTamaroff My entire day was fucking shite and I'm fed up with everything. I don't want to take oral exams because I don't give a damn what this particular lecturer thinks is an appropriate grade of my current skill.

And so on.

Jonas Teuwen

@JasperLoy Studios?

Michael Greinecker

I'M originally from Salzburg but are in Vienna since 2001. I'm moving to Innsbruck next month though.

Matt N.

So I hereby quit preparing for CA.

Jonas Teuwen

9:26 PM

Oh, I know somebody that works in Innsbruck... She will move to Zürich.

user19161

@MattN. Well, it's OK as long as you pass and get your degree.

Michael Greinecker

@PeterTamaroff With the cofinite topology. We can arrange the proof

Peter Tamaroff

@MattN. When do you have the oral exams?

Old John

@JonasTeuwen Hmm - everybody seems to be moving - I am staying right here :)

Matt N.

@PeterTamaroff CA is on the 9th.

Peter Tamaroff

9:27 PM

@MattN. You have a week still!

Don't get carried away by emotion, young Vulcan.

Matt N.

@PeterTamaroff Which I'll use to read functional analysis.

@PeterTamaroff Too late : )

Peter Tamaroff

@MattN. What is the oral exam about?

@MichaelGreinecker OK.

So I start with this

"If $\mathfrak I$ is a $T_1$ topology, then the open sets are subsets $A$ of $X$ such that $X\setminus A$ is finite.

user19161

@jonas I just started revising all my LaTeX. I hope to complete my LaTeX and PSTricks studies this year. Then next year I will do math.

Jonas Teuwen

@OldJohn 8-).

Peter Tamaroff

The "largest" of them all not considering $X$ are those of the form $X\setminus \{a\}$ with $a$ a point in the space."

Jonas Teuwen

9:29 PM

@JasperLoy Why would it take a bloody half year? 8-).

Stop doing shit so procedurally/linear. Too slow.

user19161

@JonasTeuwen Well, I mean I would do other things too. And as you know, I am very unwell now.

Matt N.

@PeterTamaroff Commutative algebra: tensor products, Ext, Tor, inverse limits, direct limits and so on...

Michael Greinecker

@PeterTamaroff Noo! A T1 topology can contain many other open sets. This only applies to the coarsest such topology, the cofinite one.

anon

@PeterTamaroff cofinite sets are necessarily open in arbitrary T1 topologies, but the converse is not true

Jonas Teuwen

@JasperLoy Yes, but that is no excuse, that is a motivation to work harder on it.

Peter Tamaroff

9:31 PM

@MichaelGreinecker Darn.

Jonas Teuwen

Small steps don't work. In Dutch we have this saying Zachte heelmeesters laten stinkende wonden..

user19161

@JonasTeuwen Exactly, but it will still take time. Anyway, the PSTricks has many, many packages you know.

anon

Perhaps you should also convince yourself that cofinite sets are always open in T1 topologies :)

Jonas Teuwen

@JasperLoy You don't need to know all of them. Just know where to find it when you need it.

user19161

@JonasTeuwen Well, I like reading phone books. :-)

Old John

9:32 PM

@JasperLoy Nah - phone books (even maths ones) are for looking things up - not for reading

Matt N.

Going now. Bye everyone.

Michael Greinecker

Bye!

user19161

@OldJohn Oh I remember you were talking about math phone books as well. On that note, I think that Lang's Algebra is a phone book as well.

user19161

@MattN. Good luck!

Peter Tamaroff

@anon You're saying that $\{a\}$ is closed and open?

Jonas Teuwen

9:33 PM

@OldJohn Yes, he send the phonebook to me.

Peter Tamaroff

I know that is true in the discrete topology.

anon

@PeterTamaroff No, I'm saying $\{a\}$ is necessarily closed, but not necessarily open.

Old John

@JasperLoy Yep - along with Doob's Potential Theory - I might give that one away too some day

Henning Makholm

@PeterTamaroff He said "cofinite", not "finite"

anon

More generally, every finite set is closed in a T1 topology.

user19161

9:34 PM

@OldJohn You gave Jonas a book?

Old John

@JasperLoy Yes - but he hasn't got it yet :)

user19161

@OldJohn May I know what book is it?

Old John

@JasperLoy Federer - geometric Measure Theory

Peter Tamaroff

@anon OK, sorry.

user19161

@OldJohn Haha, I have a copy too.

user19161

9:36 PM

@jonas Now both of us have a copy of the Federer phone book!

Michael Greinecker

@OldJohn It seems to be widely recognized at the hardest mesure theory book ever.

Jonas Teuwen

@JasperLoy Not yet!

Old John

@JasperLoy I consulted it once or twice - but never read it :)

Peter Tamaroff

@MichaelGreinecker Any maths should be difficult if explained by a tennis player.

Old John

@MichaelGreinecker Yep - I would put it in the same category as Doob's potential theory

Peter Tamaroff

9:37 PM

@anon I understand that.

user19161

@PeterTamaroff I prefer Nadal to Federer. Nadal looks cuter as well.

Old John

Now I am into number theory, so my analysis books are not much use to me

Peter Tamaroff

I just got a little confused with the open and cofinite meddle.

anon

@PeterTamaroff Can you explain why all singletons are closed, and therefore how every finite set is closed (ie every cofinite set is open) in T1 topologies?

Peter Tamaroff

@anon Yes.

If $\{a_0\}$ and $\{a_1\}$ are closed, then, since a finite union of closed sets is closed, then by induction on the cardinality of the set, any finite set of the form $\{a_0,a_1,\dots,a_{n-1}\}\sim \{0,1,\dots,n-1\}$ with $n\in \Bbb N$ is closed.

user19161

9:39 PM

I see that @peter has finally moved on to topological spaces, phew!

Old John

@JasperLoy :)

anon

@PeterTamaroff yep. and how do you get the fact that singletons are closed?

Peter Tamaroff

@anon I was given that as a defintion :P

anon

@PeterTamaroff definition of what?

Peter Tamaroff

But I also know the separation axiom of $T_1$ spaces

@anon A space is $T_1$ is every singleton is closed.

user19161

9:41 PM

OK so I checked Amazon and it says that Lee's smooth manifolds book should be out 31 Jul (second edition) but I think there will be a delay as it is still not out.

Old John

@jonas @JasperLoy but ... the other day I bought another phone book GTM - just to keep my shelves full :)

BenjaLim

mornin

user19161

@OldJohn And that is?

anon

@PeterTamaroff okay. can you derive the fact that singletons are closed from the separation axiom more commonly known as T1?

Peter Tamaroff

The $T_1$ axiom is that for evert distinct pair of points $a,b$ there is a pair of open sets $O_a,O_b$ so that $a\in O_a$ but $a\notin O_b$ and vice versa.

Old John

9:42 PM

@JasperLoy Cohen: number theory Vol 1

user19161

@OldJohn I should not have wasted money in the past on books that I hardly read. I currently have 83 on my shelves. :-)

Peter Tamaroff

@anon Yes. I just wrote the axiom up there BTW

user19161

@OldJohn Ah, not to be confused with Cohn, the algebra guy.

Peter Tamaroff

@anon One can think of singletons as the building blocks so to say.

Old John

@JasperLoy similar here :(

Jonas Teuwen

9:43 PM

@OldJohn Hmm :-).

I have... too many bookshelves.

Old John

@JasperLoy Indeed - they should be made to have dis-similar names

anon

@PeterTamaroff building blocks of the closed sets in the cofinite topology, yes.

user19161

If any of you come to visit me, I may give away some of my unwanted books which are in perfect condition!

Old John

@JasperLoy Same here - but where are you?

anon

Anyway, using the separation axiom: fix $x\in X$, and for all $a\in X$ not $=x$, let $O_a$ be an open set containing $a$ but not $x$. Then $\bigcup_{a\in X\setminus\{x\}} O_a=X\setminus\{x\}$ is a union of open sets and hence open, hence $\{x\}$ is closed.

user19161

9:45 PM

@OldJohn Singapore, your ex colony!

Old John

@JasperLoy Of course - you told me, but I fotgot

@JasperLoy @JonasTeuwen we should organise a book-exchange system on MSE - there must be loads of people who have books that could be better used by others ...

Jonas Teuwen

@OldJohn Good idea! :-).

Old John

... and people who need books they can't find

Peter Tamaroff

@anon Got it!

user19161

@OldJohn Yeah, I am fine as long as the book is in decent condition and has no markings.

Jonas Teuwen

9:47 PM

@OldJohn Or a separate website. I believe SE can give free ads for non-profit projects.

Old John

@JonasTeuwen That could be a useful approach

user19161

When I start writing my own books, I will make a website and put up the chapters one by one.

Old John

@JasperLoy excellent

user19161

I will make it like Milne's website. He has thousands of pages in notes!

Old John

I suspect that many of us also have quite large collections of notes and texts in pdf/djvu format - we ought to have a repository where we could put them all
(at least the legal ones!)

Peter Tamaroff

9:53 PM

@OldJohn Dropbox is cool for that.

anon

doesn't dropbox have like a 2gb limit?

Old John

Yeah - far too limited

Peter Tamaroff

@anon If we are lots of users we can get that expanded.

@OldJohn My DJVU files weigh in average 4-10 mb

user19161

oocities.org/alex_stef/mylist.html