Hi :-).

@JonasTeuwen I discovered that we had a leak in the garage roof and some of my old maths books have been pretty much destroyed :(

@OldJohn Oh :-(. Mushrooms?

Just damp and mould - no mushrooms. Nothing too important, fortunately - old stuff like Halmos "measure Theory"

@OldJohn They can be restored if they are really valuable to you.

If you value Halmos's book, I also have it and I don't like it so I can put it in the mail to you 8-).

@JonasTeuwen I don't really think they are worth it - I have more recent books on measure - if I ever want to look anything up, but that is a bit doubtful now - my interests are now number theoretic :)

But it was a very kind offer!

@OldJohn Okay. You can kill the mould. Store them in a clean place perhaps you want to look into them in the future.

Painting and book restaurators use this mushroom killer to preserve the books better :-).

@JonasTeuwen There was a period in my life when I probably had too much money and bought lots of books that I have never read, and probably never will now - I really ought to try to pass some of them on to people who could make use of them

@OldJohn I have the same experiences... :D.

e.g. Federer (bought in Yellow sale) - never looked it after the first day!

Oh! I want that one.

Not that I am going to read it eh.

@JonasTeuwen You are very welcome to my copy - it is useless for number theory :P

@OldJohn :P. But you might go into geometric number theory! :P. And then...

@OldJohn Am I wrong to think the Cantor set contains only the points of the form $a/3^m$?

@JonasTeuwen Nah - not at my age!

@OldJohn Why not?

@JonasTeuwen Hmm - I have too many interests - maths is only one of them

@OldJohn Ah! Whisky?

@JonasTeuwen LOL!

It is one of mine! 8-).

@PeterTamaroff There is a connection with base 3 expansions of reals in the unit interval - but it is slightly tricky - I can explain (I think) if you want

@JonasTeuwen My other interests are bonsai, photography, playing the lute (badly), languages, .....

@OldJohn Cool! Bonsai! What languages?

Do you speak Klingon?

@OldJohn Oh, go ahead, please.

@JonasTeuwen No Klingon - but Russian, Persian, Arabic, tiny bit of Mandarin, (Latin, French, German when I was at school)

@PeterTamaroff OK - first stage is to remove all numbers (inn base 3) which have a 1 after the "decimal" point - leaving just the ones starting "0.0..." and those starting "0.2..."

@OldJohn I never understood hwo to write in "decimal" and binary.

What would $0.01010111$ be in decimal, for example?

@PeterTamaroff Just think of the "column headings" in base 10 : 1, 10, 100, ... and for decimals they are 1/10, 1/100, 1/1000 ... then change them into powers of 2 or 3 instead of powers of 10

@JonasTeuwen The offer is there about Federer - if you would like it, just let me know where to send it

@OldJohn Wow, thanks! :-) Maybe trade for something, I will look into what I have and might be related to number theory!

@OldJohn Russian! I also wanted to learn Russian, but maybe first a Scandinavian language.

@JonasTeuwen OK - no problem!

@OldJohn Oh! Duh!

@JonasTeuwen My step-daughter is Russian, so my grandson is growing up bilingual English-Russian, so that is my incentive

@OldJohn I want to learn russian!

@OldJohn Ah, cool!

Scandinavian languages are hard - if you go there they always speak fluent English!

@OldJohn Yes... 8-).

@OldJohn OK, that number should then be $0.33984357$

But it is hard getting round Moscow without some Russian - but that gives an incentive to learn more

@OldJohn I once met a Finnish guy that was very good at English

@PeterTamaroff what number is that??

Finnish is one language I will never attempt!

@OldJohn $0.01010111_2$

@PeterTamaroff Ah! - OK!

@OldJohn OK, you can move on now.

@PeterTamaroff OK - so the idea is that by removing the base 3 "decimals" which have a 1 after the point, you are basically removing the middle third of the unit interval

Then we remove $0.01....$ and $0.21$ right?

leaving just the ones that start 0.0 and those that start 0.2

@OldJohn OK

Nearly!

next we remove those with a 1 in the second place

leaving 0.00... 0.02... 0.20... 0.22...

@OldJohn I get it.

You can see it visually on the wikipedia page

@OldJohn Now we're left with 000,002,020,022,200,202,220,222

Basically we remove all those numbers that have a 1 in base 3, right?

yes - all the numbers that start with those digits

@OldJohn I know how it is visually.

@OldJohn Start?

@PeterTamaroff yep

@PeterTamaroff I mean the numbers that have 0.22 in the initial 2 places

@OldJohn OK

You can use this argument to show that the Cantor set is uncountable

@OldJohn TO make a bijection between $C$ and $(0,1)$?

Yes - I think so

Hi everyone!I just came by to say that I passed in my Real analysis test,@PeterTamaroff!

@MeAndMath Congrats!

@PeterTamaroff imagine a mapping from the vase b "decimals" containing only 0s and 2s to the binary "decimals" ... you replace the 2s with 1s

@PeterTamaroff Thanks!I´m very happy!

@MeAndMath Well done!

@OldJohn Thanks ,Oldjohn!

Thanks everyone in math exchange!You helped a lot!

@MeAndMath What did you get, if I may ask?

@OldJohn Right. It is like Cantor's diagonal argument for $\Bbb R$

@PeterTamaroff 8,5 of 10,0

@PeterTamaroff No, not really

@OldJohn Well, by a diagonal argument you prove $B=\{\{a_n\}_{n\in \Bbb N}:a_n\in \{0,1\}\}$ is uncountable.

the bijection between $C$ and $\mathbb{R}$ just shows that the two sets have the same cardinality

@MeAndMath Wow, great. Congrats!

Now biject $B\to \Bbb R$ by a binary to decimal correspondence.

@PeterTamaroff OK, yes

@Gigili Thanks a lot!

@MeAndMath Does it round up to $9$ maybe? =D

@PeterTamaroff no,no,unfortunately :-P

@OldJohn So finally, the Cantor set contains exactly those real numbers in $\Bbb R$ with base 3 "decimal" expansion with no $1$s right?

@PeterTamaroff pretty much, I think, yes

In fact, yes - exactly!

@OldJohn Can you show me how to be sure of that?

@PeterTamaroff well - any number with an expansion containing a 1 somewhere would have been removed at some stage

Show than every number with only 0,2's in its ternary expansion will be in every stage of the Cantor set and thus in the Cantor set itself, and conversely show that any number with a 1 in its ternary expansion will be taken out at some stage.

@OldJohn And all the only choices we have are $0,1,2$

@PeterTamaroff yep

HI .... is this series convergent?? $$ \sum_{n=2}^{\infty} {1 \over (\ln n)^{\ln n}}$$

Now anon is here, he will provide better explanations than I can manage at this time of night :)

@anon I'm just asking because we take the intersection. However, since $C_0\supset C_1\supset C_2\cdots\supset C_n\supset\cdots$, the intersection $C=\bigcap_{n=01}^\infty C_n$ will clearly contain only those points.

\supset

@anon Thanks for the prime race article - a nice one

no prob.

Is it my imagination. or is MK producing a bunch of new questions recently?

@BillDubuque I heard you mention you studied Braid groups. I spotted an error in an answer I posted about the topic, and while I can fix it by simply changing it, I want to understand why it exists better. When you get the time could you ping me?

@anon That was over 30 years ago when I took a knot theory course from George W. Whitehead. Alas, I haven't thought much about them since.

@OldJohn Can we regard the cantor set as a topological space?

Subspace of $\Bbb R$?

Ah. Well. Maybe it will come back, because it's fairly basic - just the construction of Bn and Pn via algebraic topological considerations. In my answer here, the diagram I have on the right is inaccurate: the orange line would first cross the green, and then go to the left instead of the right.

@BrianMScott I need you!

@PeterTamaroff absolutely - any subset can be regarded as a topological space

@OldJohn Can I call an homogeneous space self similar?

Can I, can I?

Pretty please?

I could fix this by just changing the orange line on the left, but the fact remains that the diagram on the left is a perfectly valid illustration of an example. Thus it must be the case that the orange loops can be moved around until it represents a valid braid diagram. I was wondering about what sort of moves those would be and why they're justified.

Hmm- homogeneous - not sure (Can't remember) - but definitely self-similar

@OldJohn Homogeneous means given any pair of points $a,b$ in the space there exists an homeomorphism mapping $a$ to $b$

@experimentX that would diverge. Check out the Cauchy Condensation Test

That the space "looks" the same from $a$ or $b$.

@PeterTamaroff Hmm - I think you need to ask a topologist about that, I am really not sure :(

@anon I wish I could help. I'd have to find my old notes to refresh my memory after 3 decades, but they are long lost. Btw, what did you use to compose the diagrams.

thanks @robjohn i was about to post question

@BillDubuque MSPaint. :P

@experimentX Rats! I should have waited :-)

@robjohn it diverges?

@MeAndMath oops! It converges using that test.

Hmm. Does there exist an order isomorphism between $\Bbb Q\cap[0,a]$ and $\Bbb Q\cap[0,b]$ even if $a$ and $b$ are $\Bbb Q$-linearly independent? (with the usual order)

@robjohn :D

I imagine so, but you'd have to divvy an interval into sections and then expand/shrink the sections independently.

lol ... why am i not being able to evaluate this integral then?

link on WA

it got messed up

Check your parenthesis

It shows up as int(log(x)/log(x)dx) to me....

@experimentX That integral has nothing to do with the sum you posted.

not a parenth problem, a carrot ^ problem

it's here

Perhaps a ^ instead of a /?

i tried integral test before test robjohn suggested

i got no result from mathematica as well as WA

it doesn't converge

@experimentX That function probably has no closed form and Mma spits back the same integral when it can't do anything else.

@anon really? the sum converges by the Cauchy Condensation test.

hmm, I must be reasoning wrong

but ... it's a definite integral ... should Mathematica do numerical integration and give me result?

seems to converge to something just over 5

@anon Try subbing $x\mapsto e^x$

yes you're right

$\int\frac{e^x}{x^x}\mathrm{d}x$ converges

oh ..

@anon don't worry, at first glance, I said the sum diverged, as well.

oh, I don't worry.

@anon $\log$s get confusing sometimes.

still WA does not give me the result!!

@OldJohn how did you evaluate??

@experimentX I tried WA with upper limits of 100, 1000, ... 100000 - they seem to be settling to 5and a bit (this is only a rough guide!)

How do I figure out what the fuqe is issuing a \newpage or \clearpage?

i'm completely new to this type of integration ... even machines hate infinity

all right ... thanks everyone for your effort!! wasn't my problem though :D

@JonasTeuwen Your LaTeX document is mysteriously skipping a page sometimes?

@HenryT.Horton It is.

@HenryT.Horton It puts some stuff on the next page... while... there is no \clearpage in between.

Between sections?

@JasperLoy scrrprt.

@HenryT.Horton No. Between two lines...

@JasperLoy Yea, I am pretty aware of the internal workings of TeX :-).

But my document is full of bloody monkey macros.

@JasperLoy Yes, if I cannot figure out I will :-).

But I want to be one myself, so I have to figure this out.

yet another question from MK - how many is that in the last 48 hours?

Hmm some people just seem to want to push the boundaries, I think

Hmm... for some reason biblatex seems to issue a new page before and after this bibliography-thing. I'm not sure why cause there is no such command in its source 8-).

Or perhaps the file hooks.

@OldJohn who is MK?

what do the black stars next to some comments mean?

in the right side?

@ChuckFernández "Star this message as useful/interesting for the transcript"

why are some of them black and others white in the middle?

@ChuckFernández Some are pinned (super-star), and will be there for about tow weeks. Other stars are normal and will go down when new stars come!

Blistering... hot... here.

@robjohn Makoto Kato, I guess.

@JonasTeuwen What is blistering?

Cool Japanese guy is cool.

@robjohn The heat! 8-). 26.6°C.

@Gigili Thanks. I will have to look.

i thought people over here werent very fond of makoto kato

@ChuckFernández Some are pinned (super-star-ish) so they'll be on the star wall for about two weeks, others are normal stars and will go away.

@JonasTeuwen I figured it was the heat. That's 80°F. That's the temperature here when it gets cooler in the evening.

@robjohn Holy cow... is it humid? Here it very humid.

Why dont you want to answer 2012 questions?

oh wow, none of the imo questions use 2012

@JonasTeuwen here is a graph of our weather. Over the weekend it was over 104°F (40°C).

@robjohn ...warm...

@robjohn 98% humidity?

@JonasTeuwen When the temperature drops, yes.

Wow, I've capped for the first time in a long while.

@robjohn Don't go showing off you caps dude. Some people like me haven't capped in a while, you know?

I capped once in my life.

@PeterTamaroff Nor have I. Sorry about that.

I think I figured out the bug... Just delete everything until no bug is left! 8-).

@MarianoSuárez-Alvarez I just had a fun time proving some facts about the Cantor set.

Such a crazy object.

@PeterTamaroff Now go for the Fat Cantor Set :-)))).

(I'll be back later: I am not really here now :-) )

@JonasTeuwen What's that?

@PeterTamaroff Check wikipedia, it is quite cool. And now you know more about this set you can check if you really understand :-).

Smith–Volterra–Cantor set

Yeah. The fatty.

Does "Patience" refer to his or ours

@DylanMoreland The user? Yeah, that's a fun choice of a username.

yoyoyo

@DylanMoreland hey

@PeterTamaroff Yes. Maybe I'm being too harsh again.

@DylanMoreland I have Jarod Alper as my lecturer for algebra 3!!!!!!!!!!!!

Hey seems so cool

@DylanMoreland the course is on brian hall's book on lie groups

but given it's an algebra course and a lot of people don't know what a differentiable manifold is

Hi BenjaLim.

It will mainly be on Matrix lie groups

That sounds good. He could teach you a lot of algebraic geometry.

Oh, yeah, Hall's book is pretty down to earth.

Maybe almost too much so.

@DylanMoreland I know I am now looking at him as a potential honours supervisor

@DylanMoreland And he said he would be willing to do a reading course next semester on algebraic curves with me :D

@DylanMoreland Have you met him???

No. I'm just aware of a lot of people.

ah ok :D

@DylanMoreland you study number theory under Matt Emerton

We wouldn't really run in the same circles. I would like to go to more algebraic geometry conferences but there's just no time.

@DylanMoreland Yeah I don't think he's into like arithmetic geometry or anything like that

@DylanMoreland He seems like a pretty cool guy, he just landed in australia and then yesterday at class he was like "for this class you will need to know a little bit about what a manifold is and some complex analysis" and I'm like oh crap

They used to just be called ifolds until I started working with them.

Well, probably good to brush up on manifolds.

You just need to become a little comfortable with the language. That's all basic differential geometry is.

@HenryT.Horton HAHAHHAHAH

@BenjaLim Mariano teaches Differential Geoemtry.

And everything in Hall's book is inside of a matrix space so that's even easier to grasp, I think.

Haven't read much of it.

@DylanMoreland oh thanks that makes me fell better

@DylanMoreland I have studied a little bit about representations of finite groups

some of the small dihedral and symmetric grouops

@PeterTamaroff Did you invade his office?

@BenjaLim Not yet!

you should

Here is a cute problem: $\displaystyle\sum_{k=2}^\infty2^{-k}\tan(2^{-k}\pi)$

@DylanMoreland you study number theory under matthew emerton ?

Is his australian accent strong?

Yes.

ahhahahahahah

@BenjaLim Do you watch Wilfred?

@DylanMoreland jarod was saying yesterday how we can identify $GL_n(\Bbb{R})$ with $\Bbb{R}^{2n}$

But what is the topology on the group?

It's an open subset of $\mathbb R^{n^2}$, you mean.

Then it's just the subspace topology.

Of $\Bbb R^{n^2}$, you mean