12:04 PM
@BenjaLim Hi.
@ami Sure, send it to me. If you arrive at such an equation I would rather check the method instead of trying to solve this.

@JonasTeuwen Can I ask you something in analysis?
If we have a continuous function $f : K \times L \rightarrow \Bbb{R}$

@BenjaLim Sure, you can always try.

where $K,L$ are compact subsets of $\Bbb{R}^n$
Then the function $m(x) = \max_{y \in L} f(x,y)$ I have proven to be uniformly continuous
@JonasTeuwen Now does this help me in proving that $$\max_{x \in K} m(x) = \max_{x \in K,y \in L} f(x,y)$$?

So basically you are trying to prove that the maximum can be made "in one"?

yeah

12:08 PM
So, one inequality should be obvious right? That's usually the case 8-).
In any case, don't try to prove $=$ directly but do $\leqslant$ and $\geqslant$.

I think it is clear that R.H.S. \geq L.H.S.
Because that's the max taken over the whole set :D

Hmm, but simultaneously?

yes why not?

You have $m(x) \geqslant f(x, y)$.

ok
@JonasTeuwen ok

12:11 PM
Darn! the LaTeX support in MathJax page seems to be broken.

@JonasTeuwen does uniform continuity help?

Actually, most of the links from the main MathJax site seem to be broken.

@robjohn Did you see the problem I put above?

@BenjaLim I would say that $$\max_{x \in K} m(x) \ge \max_{x \in K,y \in L} f(x,y)$$ but that is all because we don't know that $f$ is onto.

ok
hmmm
Somehow I need to use uniform continuity

12:20 PM
Actually, we have $$\max_{x \in K} m(x) \ge \max_{x \in K,y \in L} m(f(x,y))$$

ok
hmmmm

@BenjaLim what is $m$?

@robjohn $m = \max_{y \in L} f(x,y)$

@BenjaLim Ah, okay, then my first statement was correct, by chance.

@robjohn K,L compact

12:23 PM
@BenjaLim hang on, I am looking back at your question, knowing what $m$ is.

@robjohn hmmm....
Funny because I don't get how uniform continuity comes into the picture....

I don't think it does for this. However, you are trying to show$$\max_{x\in K}\left(\max_{y\in L}f(x,y)\right)=\max_{(x,y)\in K\times L}f(x,y)$$
right?

wait is that what I'm trying to show?
I'm not sure if your formulation above is equivalent to mine....
YES

the stuff in the parens is $m(x)$

yup
that's what I'm trying to do :D
You see the reason why I think uniform continuity comes in
is because this relies on the compactness of both $K$ and $L$

12:31 PM
@BenjaLim The fact that you can get a max rather than just a sup is dependent on the compactness of $K$ and $L$

yeah but there must be something more to it....
I mean I got the uniform continuity from like $\epsilon - \delta$ arguments and compactness

It is obvious that the left side is $\le$ the right side, yes?

you mean greater?
@robjohn because $m(x) \geq f(x,y)$ for all $y$

@BenjaLim Sorry, colleague walked in.

and so taking the max over $K$

12:33 PM
no, $\le$ since it is a max over a possibly smaller set

@J.M. certainly :)

How is that possible @robjohn?

on the left we are looking at one value of y for each x

You are saying that $\max_{y \in L} f(x,y) \leq f(x,y)$?

@BenjaLim how is what possible?

12:35 PM

Hopefully it's both $\le$ and $\ge$, right? :P

It is pretty obvious that $m(x) \geqslant f(x, y)$ right?

yes jonas
but rob is saying the opposite @JonasTeuwen @robjohn

@CLarue yes :-)

12:47 PM
First of all,
$$\max_{x\in K}\left(\max_{y\in L}f(x,y)\right)\le\max_{(x,y)\in K\times L}f(x,y)$$
because for each $x\in K$, we are picking one value of $y_x\in L$ on the left whereas we are taking all possible pairs on the right. That is, the set of $(x,y)$ we are maximizing over on the left is smaller than the set of $(x,y)$ on the right.

However,
$$\max_{x\in K}\left(\max_{y\in L}f(x,y)\right)\ge\max_{(x,y)\in K\times L}f(x,y)$$
because for the $(x_0,y_0)$ where the max is attained on the right, we have that
Put the two together...

@robjohn :D
thanks

@BenjaLim You were concentrating on the second part, and I was trying to talk about the first part.

I'll try to do it myself now @robjohn
@robjohn Actually I don't understand the second inequality.....

@BenjaLim but that was the one you were arguing for before.

I'm stupid.
for every $y \in L$ we have that $m(x) \geq f(x,y)$

12:57 PM
$$\max_{x\in K}\left(\max_{y\in L}f(x,y)\right)\ge f(x_0,y_0)=\max_{(x,y)\in K\times L}f(x,y)$$

@robjohn I have for each $y \in L$ that $m(x) \geq f(x,y)$

Put $x_0$ in for $x$

yes and so in particular for $y_0 \in L$ and $x_0 \in K$ that $m(x_0) \geq f(x_0,y_0)$

yes

$\max_{x \in K} m(x) \geq m(x_0) \geq f(x_0,y_0)$
@robjohn YES!!

1:00 PM
good

@robjohn Hmmm that uniform continuity thing really threw me off

It doesn't affect anything

yeah that was a red herring
@robjohn Actually showing this equality was more easy than the uniform continuity...

8-).

@JonasTeuwen My analysis is not very good.
Actually for the topological stuff I'm ok

1:01 PM
@BenjaLim But soon... it will!

@BenjaLim yeah, this is pretty much a no-brainer when you understand what is going on.

when we start talking about derivatives I'm useless.

@JonasTeuwen :D

1:14 PM
@robjohn I'm going to bed now
thanks man!!!
bye @JonasTeuwen

@BenjaLim: I think this answer is similar in nature to what you were dealing with.

@BenjaLim Bye bye :-).

at least that $\displaystyle A\subset B\Rightarrow\sup\limits_{x\in A}f(x)\le\sup\limits_{x\in B}f(x)$

1:34 PM
@BillDubuque: that didn't seem offensive to me. Did the flag get thrown out?

HI folks

Hi :-).
Do you guys know the software called Mendeley (and Zotero)? It is pretty cool as a reference manager. The Bibtex export seems to suck slightly.
@OldJohn Damn! Missed it 8-).

@JonasTeuwen I was just asking what it did - and by the time I typed it, you had answered ;)

Excellent.

off to the park. bbl

1:49 PM
@robjohn Have fun!

@robjohn Enjoy!

2:02 PM
It's said that $\Bbb R^+$ could be well-ordered in some order $\prec$ in Don Knuth's The Art of Computer Programming.
In ZFC axiomatic system.

@FrankScience Yes, but don't expect to be able to write down an explicit well-ordering

@OldJohn Why?

I knew you were going to ask that!
Not sure I can recall exactly why, but vaguely remember something to do with the axiom of choice

Think it's because there is a bijection between $\Bbb{R}$ and some cardinal, which by AC, is an ordinal.

Just out of curiosity.
I read it in TAOCP where Don Knuth declares that nobody could point it out explicitly.
I have no idea about AC.

2:11 PM
Do you know about the construction of the Vitali set, in dealing with the Lebesgue measure on the real numbers? Because the reason for that not being explicit is the same here.

2:57 PM
why is it latex doesnt work on se sometimes?
instead of going to matah mode it put a square around the text

@ChuckFernández That usually means that there is an error in the TeX.

ok
to make fractions we use \frac right?

Yep.

$\frac{1}{2}$
kk

Beware that backslashes can cause errors on the site that would not be errors for a LaTeX compiler on your machine. I think there's some conflict.

3:14 PM
what do you mean?
what can i use instead of backslashes?

Use backslashes! But sometimes you have to use extras.
Like, for example, open up a draft of a new question on the main site and type in $a \, b$. That won't do what it should.
But \$a \\, b \$ does.
Something to do with Markdown escaping, I think.

I feel like a monster. A new user posted a "this should be a comment" answer and I flagged it to be made into a comment... he might lose his first two upvotes
I didn't want it to happen this way.

@HenryT.Horton Yeah I never know what to do... certainly I had a few such answers early on which never got converted and it helped me reach the comment threshold, which was nice.
On the other hand, reaching 50 is not so hard. There's a lot of low-hanging fruit.

4:03 PM
Hi

I am sad because I thought I had invented a topological space with a really interesting weird property and then I realized that my whole idea was completely wrong.
I weep for my lost pathology.

the number of coinfinite subsets of Z is continuum right?

Yes.

yes, duh: the power set of the subset of even integers is continuum by itself
why do we need CH?

Who said we need CH?
Certainly not me.
I get correct answers by an iterative process that eventually approaches within $\epsilon$ of the truth.

4:17 PM
Why do we need math?

Without math we would have nothing to do in the bath tub.

it's tautological that one needs math to do math!

@MarkDominus I don't have bath tub.
@anon But isn't all correct math a tautology?

hello there my mathematician friends
I have a quick question for you and it will sound very stupid

First year high school?

4:22 PM
there is a famous knot theorist named Louis Kauffman. So close.

yes.

@anon I know a mathematician called "Frank Sommen". Sommen is "sums" in Dutch.

lol

So I once told him: "That's a good name for a mathematician." He ignored me.

now my question is
do you first solve the distributivity
or devide 48 by 2?

4:23 PM
that's not a question about math itself, that's mathematical conventions

Then why do you say yes?
@anon But a very common one...

unfortunately

so what does the convention state?
I believe it's two

convention states you don't deliberately write ambiguous expressions
but if you want, you can look up PEMDAS

ah yes PEMDAS
so multiplication before division?

4:25 PM
I think they're supposed to happen at the same stage. Whatever that means.
I would never write an expression in this way, so I don't know how to answer the question, sorry. Many similar questions have been asked on the main site.

@anon You fancy (reverse) Polish more eh?

anyway thank you for your time gentlemen ^^
I shall now crawl back into my basement

@LucasKauffman If you look at the discussion on Wikipedia I think that strictly applying the rules leads one to interpret this as $(48/2)(9 + 3)$. I don't quite see why, but it jives with their example.

Louis Kauffman's On Knots book was one of the things I read as an undergraduate. It it a really strange book.

I guess it's like subtraction. When you see $/x$ you replace it with $* (1/x)$ and get on with life.
But I would recommend avoiding this whenever possible. The fact that there are memes and questions about this should be a clue that it isn't worth whatever space you're saving.

4:30 PM
:p

I think an invisible multiplication sign binds tighter than a visible one. I would understand $48÷2(9+3)$ differently than $48÷2·(9+3)$.

@MarkDominus The what?? The visibility of an operation is not a consideration to the order of operations.

The symbol used for an operation is a consideration.

@MarkDominus In the order they're performed? No.

Consider for example $a/bc$ versus $a/b·c$. The first one is certainly $a/(bc)$, not $(a/b)c$. The second one is possibly ambiguous, but much more likely to be $(a/b)c$ than $a/(bc)$.
That is why $\TeX$ typesets more space around the $\cdot$ than around the $/$.
Do you want to claim that $a/bc = (a/b)c$ because of strict left-to-right evaluation of multiplication and division? If so, I think you are on very shaky ground.

4:44 PM
Mmm... is the $a/b$ actually meant be written this way in $\LaTeX$?

Meant to be written what way? Are you asking if Donald Knuth knows that people use $/$ to indicate division? The answer to that is yes.

@MarkDominus Ambiguous notation does not modify order of operations, it's simply ambiguous, there is no proper order. The picture has no ambiguous notation either.

@MarkDominus If it is meant to be typeset this way. It is a primitive of the keyboard so to speak.
You do have the package \xfrac.

@ChrisS So you do claim that $a/bc$ means the same as ${a/b}c$ then?

Perhaps $^a/_b$.

4:46 PM
@MarkDominus Nope. I claim that "a/bc" is ambiguous, and there is therefor no strict order of operations possible.

@MarkDominus How many people use $\TeX$ other than say Knuth himself?
@ChrisS But isn't that a non-issue? That is not how you would write it on paper but in a programming language. And they have their own conventions.

@ChrisS Okay, you are mistaken, as perusal of many books on mathematics will clearly show.
$a/bc$ is widely used and invariably means $a/(bc)$.

Mistaken... In what way? That is not the notation you would see in a math book.

@JonasTeuwen True, but the order of operations ist still not modified by the particular symbol used, which was my point.

You would see something like $^a/_b c$.

4:48 PM
@MarkDominus Actually, $a / bc$ usually means $\frac{a}{bc}$.

Sorry, yes.

@ChrisS Hmm, perhaps you know more about this but if you do stuff like that in a programming language, can it not even depend on which compiler you use how it will be seen?

I got my claims exactly backwards in the face of Chris S's astounding claim that it was ambiguous.

I also think it is ambiguous, I have never seen it written like this.
I would be confused on how to read it without understanding what they are trying to say.

Witness any algebra text that writes $\mathbb{Z} / n \mathbb{Z}$ instead of $\mathbb{Z} / (n \mathbb{Z})$, say... (though, admittedly, the other parsing is meaningless)

4:50 PM
What is a math book that uses this notation uh?

@JonasTeuwen A particular language will have syntax that defines how operators are interpreted. While syntax is largely based on mathematical notation, it's not the same. Within each language, various compilers should all interpret the syntax the same way.

@ZhenLin But that is not a number division... well if you're really nitpicky you might say so.
@ChrisS Yep, I know. But they might handle "bad syntax" differently. They might accept the same syntax, but perhaps bad syntax is parsed as well. And you don't know what is going on there, unless that is also in the description... (like with HTML).

@JonasTeuwen It should reject bad syntax.... Though you may mean that the programmer coded something different than what they intended to code and ended up with valid syntax which performs an operation differently than intended. In that case, the flawed code has no defined operation (ie computers don't read minds =] )

@ChrisS Yes, it should reject bad syntax. But some parsers as in HTML actually do also parse bad syntax as you know with IE and such... 8-).
Anyway, I am curious where the bloody monkey you would see $a/bc$ in a math book.

@JonasTeuwen You shouldn't, as it's ambiguous. ;]

4:55 PM
If I would have to grade a report like that written in LaTeX I would encircle it and write "wa da..." next to it.
Perhaps, there is a convention on how to read that but I am unaware of it. And Zhen Lin's example is really something different. There is no ambiguity because the other thing doesn't make any sense and it is a very common construction...
And is that really true that all compilers would reject syntax that is not "standard"? I doubt it. I can remember from a long long time ago that two C compilers gave different results 8-).

ha, they're threatening to downgrade Germany's credit rating...

@ZhenLin Yep. But the arguments are not new.

The end is nigh!

Hey, would you say it is better to post a fairly complicated question to math.SE or to MO?

Depends... what you call "fairly complicated".

5:09 PM
I looked in several books at random and was not able to find nearly as many examples of $a/bc$ as I thought I was. i did find one example, which you can see in the proof of theorem 4.2 here.

It is probably safer to put it on a place where the level would be too low than the other way :-).
@MarkDominus Yes, but there cannot be any confusion how to read that.

@JonasTeuwen Fairly complicated as in I talked to a fairly knowledgeable grad student in the same field did not know the answer, or if it had been answered.

The end of the proof says $v_n = u_n/n\alpha_n$.
I claim that there is never any confusion about how to read $a/bc$.
You and Chris S. are the ones claiming that the construction is "ambiguous".

@MarkDominus Yes, but then you assume intelligence. In that case I agree.
How would a computer parse that?
@AlexBecker Mm, I would try MSE perhaps, if you think there might be someone that could answer.

It's not ambiguous, we've just been arguing an hour over its meaning...

5:11 PM
Who cares? I was not discussing computers. I was talking about how the symbols are used by mathematicians.

@MarkDominus Then we were talking about something else, sorry 8-).

O_0

@anon Hmm... what is the difference eh? Ambiguous means "can be interpreted in different ways", no?
For example, in NL one could write: $\frac152$.
How would you read that? In Belgium they would read it in a different way...

@JonasTeuwen I actually already posted it to MO, because I've seen many more questions relating to the same subject there than here, but was just second-guessing myself. This is what I'm referring to.

Can I read it as "hurr durr"?

5:14 PM
@JonasTeuwen You said that if you had to grade a report with $a/bc$ you would circle it and write "wa da". What does this have to do with computer parsing?

@AlexBecker Enough little known words for me to warrant MO 8-).
@MarkDominus Nothing, but I talked about a math book before and then if someone would write it this way while you have \frac...

@JonasTeuwen Yes, I decided to post it there after noticing MSE doesn't even have a (teichmuller-theory) tag.

I might have been mixing up two lines. I was thinking about numbers and not about variables. In the case of variables, sure.
$\frac123$ is in NL $\frac12 + 3$ and Belgium $\frac32$.

Still, $a/bc$ was much less common than I imagined. In most cases where / was used, the denominator had only a single factor. This suggests to me that mathematicians in general are not as comfortable with $a/bc$ as I thought they were.

Yes, when you gave that example I remembered that I actually have seen it, but if you understand what is going on there is hardly any confusion possible.
Perhaps it does not have to do with the fact if it is comfortable or not, it is just pretty ugly.

5:23 PM
Oh, here is a nice example: page 3 of Rudin's Real and Complex Analysis uses the expression $y/2\pi$.
According to Chris S., this unambiguously means the product of $y/2$ with $\pi$.
:)

Well, I did not mention that it would be unambiguously, I was just stating that I have never seen conventions for this... so more like common sense.

On page 88, however, he writes $1/(2\pi)$.

I would call it abuse of notation. But nothing wrong with that, it would be too cumbersome if you can pretty much assume everybody knows what you mean.

The bottom of page 99 has $\delta = \eta/2k$. I will stop generating examples now.

8-). But the page 88 already... somewhat says it can be ambiguous.
Although, you would write $\pi/2$ otherwise.
Oh, right, Ilya has my book.
@MarkDominus Mmm, the quote in your profile. Don't like philosophy? 8-).

user19161
5:47 PM
@anon Oh, some call it BODMAS, interestingly.

user19161
@JonasTeuwen Reverse Polish Notation is what they use in PSTricks.

@JasperLoy I know. Postscript eh. Easy stackin'.

user19161
@JonasTeuwen Supposedly he uses Metapost for all his diagrams too. It is still being developed today.

@JasperLoy Yes... Oh well. :-).

user19161
@JonasTeuwen I'm still deciding whether to learn pgf or pstricks or xypic completely.

5:51 PM
@JasperLoy Why not just TikZ?
I would not learn something completely which is basically postscript based.
Unless your goal is purely intellectual. If it is practical: TikZ.

user19161
@JonasTeuwen Your words have added weight to the importance of pgf. But you see, it is difficult to do 3D drawings or arithmetic with pgf.

@JasperLoy The right tool for the job... Simple 3D drawings should be okay.

user19161
@JonasTeuwen Hmm, I think I will learn pgf then! Are you working on some research now?

Yes, but I have trouble concentrating.

user19161
@JonasTeuwen OK, you can tell me when you know about what you mentioned, if you want.

5:57 PM
I will :-). But I am unable to get a hold of the correct people to get more information eh.

user19161
I was contemplating whether or not to run, but I think I shan't.

Cool!

user19161
@gustavo Nice new avatar!

@JasperLoy What... is it?

We are about to leave for Mammoth Lakes. We'll be on the road for about 6 hours. See you when we get there.

user19161
6:01 PM
@JonasTeuwen I don't know. Looks like some superhero.

user19161

@robjohn Have a great time

user19161
@JonasTeuwen Oh, I know that one.

I am going for permanent head damage.

user19161
6:05 PM
There is a whole meta post on ELU about whether it is alright to use the term grammar nazi on the site.

Hoo boy.

user19161
As always, there are various different opinions.

Why exactly is it called a "grammar nazi"?
It is just very strict. Perhaps "grammar mathematician" is more suitable.

"Nazi" in colloquial American usage refers to someone who is strict and inflexible about applying rules in every possible situation.
I consider it an unfortunate usage.

Ahhh... Befehl ist Befehl.

user19161
6:06 PM
I understand it can upset people who have been through the war though.

The Neurenberg nazis.
@JasperLoy Why...? How many of them would there still be that are on ELU?
It is quite some hyperbole, and so belongs to ELU 8-)))).

user19161
@JonasTeuwen Hmm, you are right. But still...

@Jonas the objection to the term is that the real Nazis committed great evils which are obscured by reducing them to mere bureaucrats.

@MarkDominus I understand, but that is a common language construction to do such things. Exaggerate something greatly to express your opinion.

Also, people who are just arguing points of grammar might not want to be likened to Nazis, who committed genocide.

user19161
6:09 PM
But I think people should not be offended but should have a greater sense of humour.

I am not arguing the point myself, only trying to explain it to you.

@MarkDominus Heh, that I can agree with.
But for me it would feel way different to have a direct comparison rather than having the word used as some weight.

user19161
I find it weird that people apply rules unevenly. There are some users (I shall not name) who say things to upset others but act as if they did nothing. Then others say something similar and they get upset and flag them.

Oh well. It's a bitch. Whatever.

For each se.math question that has received at least 10 upvotes, let $t_i$ be the time at which the question received its $i$th upvote, and $t_0$ be the time at which it was first posted. For which question is $t_{10}-t_9 \over t_9 - t_0$ maximum?

user19161
6:11 PM
@JonasTeuwen Yeah, have a beer!

@MarkDominus Mm, do you really want an answer? Do the tools at Data.SE not suffice for this?
@JasperLoy Perhaps I should go home.

user19161
@JonasTeuwen Yes, go back and rest then if you are in the office now.

@MarkDominus Heh, so at least I can also say things that can be... coherent.
@JasperLoy Right. But this place has an airconditioning! Anyway, see you later!

user19161
@JonasTeuwen OK.

6:39 PM
Done!

@JasperLoy Yep. I made it on Cinema 4D
@JasperLoy It's called PseudoFace

Bleh 30.8°C.

Hi ...
i have have three different colored balls .. 3 red 3 green 4 yellow ... in how many ways can you arrange them!! what should i google for?
the same colored balls are indentical from each other

Binomial or multinomial coefficient. Continue from there.

oh .. than you

6:53 PM
@experimentX The answer would be $10!$ if they were all distinguishable ...
then divide by the number of ways of shuffling the reds amongst each other
(and similarly for the other colours)

Oo
looks like i did not identify the problem correctly.
@OldJohn the original problem is stated as