It looks like Tim Gowers wants to destruct Elsevier 8-).

@Srivatsan: I amended this answer to show that $\log(\Gamma(x))$ is strictly log-convex. That way I can refer to it :-)

Ha, "strictly speaking" =)

I can't read it right now, I'll do so in sometime.. Thanks for the update.

Wait: where you are going to refer to it?

"At some point, there might be nothing left to do except to roll up your sleeves, brew some coffee, and tell your graduate student to read the paper and report back to you."

Aw, it sounded familiar.

@Srivatsan In my answer.

Oh, I haven't read this question yet. I knew of its existence only yesterday/today.

facepalm I lost track of who asked which questions.

@robjohn That's how many answers you write =)

@Srivatsan I've not written many answers recently. Work has been a harsh mistress.

I'd like to add a commutative diagram to some answers I left incomplete this morning. Can some one point out, where I can draw a Commutative diagram?

@Srivatsan ?

meta: How to draw a commutative diagram?

Do you know of ...?

@MartinSleziak Oh, Thank You!

Anyway, I would probably prefer to prepare the commutative diagram locally in LaTeX (I use xypic), convert it to picture and add to the answer.

You'll get nicer diagram that way.

I don't know how to do it on LaTeX either! I am fit-for-nothing when it comes to this! @Martin

This question actually has a very interesting idea behind it. If we define $\Lambda(n)$ as the product of all primitive roots modulo $n$ (lifted up to Z before multiplication), then (a) what is the relation between $\Lambda(2p^r)$ and $\Lambda(p^r)$, (b) can $\Lambda$ be extended in a natural way to $n$ with $Z_n^\times$ noncyclic? ...

... (contd) (c) is there any special convergence or limiting function induced in the $p$-adics? (d) what are the asymptotics of $\Lambda$ and what does $v_p(\Lambda(p^r)-1)$ look like?

@KannappanSampath Well, if you plan to use commutative diagrams often, you could choose a LaTeX package you like and learn to use it. (There are several packages for drawing CDs.)

@MartinSleziak It doesn't render? There's a big box!

@KannappanSampath What does not render?

@MartinSleziak The diagram in my answer!

@anon What is $f(x,p)$ when it is not $1$? :-)

@KannappanSampath And how did you try to create it?

@robjohn: It doesn't really matter, does it?

You're missing backslash in \end{array}.

@KannappanSampath Missing a backslash before the end{}

@MartinSleziak :-)

oh wow

Ah, I see!

@anon That's why the smiley :-)

Oh. Well then it takes on the value of "off to the grocery store to go shopping" when it's not 1.

Ouch! another short blackout/brownout

There are high winds here today, so that is probably why.

@anon Thanks! that was really bothering me :-)

@JonasTeuwen It seems that Elsevier already has kind of bad reputation, if you have a look at Wikipedia article. E.g. this: At the end of 2003, the entire editorial board of the Journal of Algorithms resigned to start ACM Transactions on Algorithms with a different, lower priced publisher, at the suggestion of Journal of Algorithms founder Donald Knuth.

@MartinSleziak There is another issue with Journal of Topology as well, IIRC!

@KannappanSampath Yes, they mention it there too.

@all I would appreciste your comments on this of mine!

@Martin Look at the commutative diagram as well above!

@anon Thank You @anon.

This is the kind of criticism I like! My writing style improve a lot!

@MartinSleziak DEK wrote a long open letter to J. of Algorithms.

See this.

Thanks for the link.

Holy Shizola

@Srivatsan Are we expected to read this or scroll through? My goodness, he has the patience to write this long stuff against some one!

@robjohn Are you interested in what I found out about math reform sir?

@Skullpatrol what math reform?

@robjohn In education in the U.S.

@MartinSleziak Oops! "I'm sending copies of this letter to several friends who are interested in journal publishing but are not members of our board. But this is not an "open letter"; I would prefer not to have my remarks circulated widely. I'm emphatically not a revolutionary. I just want our journal to do the right thing."

@KannappanSampath Well, it is pretty interesting really. I haven't read it myself (because I don't have all the required historic context)

@robjohn I'm thinking of introducing Skew Math.

@anon Because it depends on your set theory?

yes

@robjohn books.google.ca/…

@anon It’s not too surprising when you start dealing with $\mathbb{R}$ and its close relatives.

@Skullpatrol can't read that page.

In my first paper I showed that the existence of a certain kind of countable decomposition of $\mathbb{R}$ is equivalent to the continuum hypothesis and therefore depends on your set theory.

@Srivatsan what would be in Skew Math ?

@robjohn No idea. The proposal is just a few minutes old now.

Only the name has been decided. :=)

@Srivatsan Cool name :-)

But you should surely know what I am alluding to... Right?

@Srivatsan Pretend I am dense and have no clue :-)

@robjohn How about this one sir? books.google.ca/…

@robjohn I thought it rhymes with New Math.

Domains hate me....

@Srivatsan Ah, I was completely dense!

@unNaturhal Domain of a function?

@Srivatsan Yes :/

@unNaturhal I think .org hates you most :-p

@robjohn Alright. :-) Maybe the reference was obscure.

@robjohn, Umh? What does it means?

@Srivatsan Perhaps if I had said it aloud...

@unNaturhal .org, .com, .au, .uk, these are all internet domains

@robjohn xD

A'right. See you later.

Have someone a bit of time to help me in finding error of this f..in' domain?

@unNaturhal Yes I do!

@Srivatsan Bye! Catch you later!

@Srivatsan See you later !

@unNaturhal Fire!

@unNaturhal what domain?

Looks like the old one! But still let me tell you:

And if you give me two seconds I write also which condition I set up in the system

Yes is the same that I asked for in the main site

OK take your time!

There comes @Chaz

There are no real roots of $x^2-4x+5$, so we don't need to worry about that square root

But despite your advice, I dont' match :

:/

@unNaturhal ??? (confused)

Remember, last time I pointed out about that $\cup$ thing! It should be $\cap$.

In the fourth condition?

@unNaturhal There shouldn't be absolutes in the 4th

Yes, in the 4th condition; $\arccos t$ makes sense, iff $|t| \le 1$.

Why? Is the argument of the $\arccos$, and must be set betwen -1 and 1 ($-1\le x\le1$)

@robjohn In which case do you mean this way: $\lvert \dfrac{2-x}{x+3}\rvert \le 1$

@KannappanSampath That is true, but if you have both, you don't need the absolutes

(i.e) without the first set contributing to the union!

@unNaturhal Yes!

$\displaystyle\frac{2-x}{x+3}\ge-1\cap\frac{2-x}{x+3}\le1$

Ah....

@robjohn Perfect!

@unNaturhal Note the $\cap$!

@robjohn Basically the math reformers are saying that students should send most of their time on understanding math while the traditionalists are saying practicing methods should be the focus.

@AsafKaragila: I saw someone else using "whatnot" without trade mark sign.

@Skullpatrol As usual, the truth lies between the extremes.

Hey there, I have two questions =\

@Skullpatrol And they’re both wrong.

So, $\bigvee$ is different than $\bigcup$?

@unNaturhal Depends on the context!

Set $S_n=\{1,2,\cdots,n\}$.

it's crowded here now, I'll ask tomorrow.

If you want, I could make a photo of what I wrote

@Gigili Go ahead and ask.

$\sum_{i=0}^{n \bigvee m}{blah}$

@robjohn Apparently the traditionalists have declared war on the reformers.

@unNaturhal Yes. There are certain contexts in which $\bigvee$ interprets to $\bigcup$, but in general they’re distinct.

$\sum_{S_n \cup S_m}{blah}$

How many ways are there to order a,b,c,d,e,e,e,e,e with no 'e' next to another 'e'?

$4!$

ways!

@robjohn If only you could see the link :-(

@Gigili

I thought I could solve it $9!/5!-5!$

@Gigili Start by arranging the five $e$’s in a row. You have four letters left, and you have to put one of them in each space between $e$’s, so as Kannappan said, there are as many ways as there are ways to arrange $a,b,c,d$ in a row, namely, $4!$.

@KannappanSampath Yes, the book agrees with you .. but I wonder why my solution doesn't work

@BrianMScott Can you see the link sir?

@Skullpatrol Which link is that?

why do you think it does work, gigili?

@Gigili Did you see what Brian has in store for you?

9!/5! certainly counts the number of ways to arrange them total, but 5! does not count the number of invalid arrangements

Yes, without the condition it'd be 9!/5!

@Skullpatrol Ah, no; I tried, but it was a forbidden page.

and 5! is when 'e's are next to each other

Am I wrong?

@Gigili exexexexe where each x is one of a, b, c, or d

@KannappanSampath, So? :/

@Gigili Does that look like all the possible arrangements allowed?

@Gigili Yes. There are lots of different ways in which at least two $e$’s can be adjacent.

Gigili, 5! counts the arrangements with all the e's put together as one big letter. But that isn't the total number of invalid arrangements: what about something like eeaeebecd?

@unNaturhal I don't understand what you mean ! As it stands, what you have written is right. Note that a $|.|$ is always positive. So, $|.| \ge -1$ is a trivial bound and you can ignore.

Ah now I see. Got it.

Thank you, you're perfect.

drive-by tutoring

@KannappanSampath, Yes, I set it as verified $\forall x \in\mathbb{R}$

@robjohn Seems to have worked.

@BrianMScott :-) Indeed.

@robjohn A few years ago we started a walk-in tutoring centre, staffed by grad assistants and a few good undergrads. It does a lot of drive-by tutoring, and it seems to be pretty effective.

@unNaturhal You mean you wrote down that the solution for that inequality is the whole of real line. So, when you intersect finally, you'll have no issues!

That's fine!

@BrianMScott It's a good idea.

Relatively inexpensive, too!

@unNaturhal I'll be happy if I knew which!

$\left|\frac{2-x}{x+3}\right|\le1$

@BrianMScott mathematicallycorrect.com Is leading the war against the reformers.

@unNaturhal I believe your simplification is correct.

@Skullpatrol Oy. It’ll take a while to wade through even the high spots.

@Skullpatrol Thank you for bringing this site to my attention! All my (not-very long) life I have battled the so called whole math.

@unNaturhal Did you check that up properly? I believe the range is much narrower!

In my lifetime I’ve seen two full cycles of drill $\to$ concepts $\to$ drill $\to$ concepts.

@FortuonPaendrag Whole Math: The current revolution in mathematics curriculum, akin to the Whole Language experiment, that emphasizes group discussion, essays, calculators and guessing and de-emphasizes basic skills and direct instruction

@unNaturhal There are two inequalities hidden there. The "$\leq 1$" will tell you $x \in \left[\dfrac{-1-\sqrt 21}{2},\dfrac{-1+\sqrt 21}{2}\right]$ and "$\geq -1$" will tell you $x \in \left(-\infty,\dfrac{-1-\sqrt 29}{2}\right] \cup \left[\dfrac{-1+\sqrt 29}{2},\infty\right)$

Now the solution is the intersection.

Note: \sqrt 20$\to\sqrt 20$ whereas \sqrt{20}$\to\sqrt{20}$.

@Skullpatrol Well, I have never found those methods described helpful in augmenting my math skill set or understanding concepts. What do you think?

@KannappanSampath, All these things are relative to the 4th condition? D:

@unNaturhal Yes

@anon Thank you anon!

@FortuonPaendrag I don't know....

@KannappanSampath, O_O

If you trust me, these are the photos: mediafire.com/file/eeezefv1qk89kyz/FuckinDomain.rar

@unNaturhal My god, 9MB file hurts! =) (Just joking!!)

@FortuonPaendrag If you click the "New-New Math" link at the top mathematicallycorrect.com/glossary.htm you'll see the definition I sent you.

There are numbers near each condition. Each number is repeated below and there are the steps to solve the inequality

@Skullpatrol Thank you, but I saw that. I was merely curious as to which side of this debate you were leaning towards..

@unNaturhal Can I like take 3 minutes off? I am writing an answer on the main site. Please bear with me!

@FortuonPaendrag I believe mathematicians should decide how their subject is taught in schools AKA "A Mathematician's Lament."

@KannappanSampath, Don't worry, I'm here ;)

@Skullpatrol Lockharts essay you mean?

@FortuonPaendrag Yes

@robjohn Which has been your most influential math book? (I want to read it too :-))

@Skullpatrol To a considerable extent they did with the New Math of the 60s. In concept it was excellent. Unfortunately, far too many schoolteachers were ill-equipped to teach it as it was intended to be taught.

I honestly doubt that they’ve anything to do with it. I’ve had the same thing happen with a pretty random selection of books, especially when I’ve been looking at a lot of previews.

@JonasTeuwen Are you not scared of the answer? It might be one that makes you want to stab yourself in the guts.

@BrianMScott In the book the author does say he has been threatened by them.

@Skullpatrol Without knowing any details, let alone the truth of the matter, I’m inclined to reserve judgement.

@BrianMScott So, Brian, which side of this issue do you lean towards ,or do you hold both in equal contempt/unconcern ?

@unNaturhal I've understood what your mistake is!

Do you know the method of wavy curves?

@FortuonPaendrag I dislike both extremes. The ideas are very important; so is a decent level of manipulative skill. And to make things more difficult, the proper balance (in terms of time spent) isn’t the same for every child.

@KannappanSampath Never seen before D:

The "method of wavy curves"... that sounds like a method my mom taught me in washing smooth surfaces.

@unNaturhal See, it's no big deal! The way you deal with condition 4 is odd! How'd you go about it:

$|\frac{2-x}{x-3}| \le 1$

@unNaturhal Oh shit, I suddenly realize you're right!

@KannappanSampath O_O

I was thinking of a $1/2$ in place of $1$.

@mk Are you serious about the comment to my answer?

Or you wanted me to add that trivial point as well?

@BrianMScott Would you agree that a mathematician is the best person to decide how the subject should be taught sir?

@Kannappan: I don't see how it's necessarily true. $H_1$ and $H_2$ could be non-abelian, and we don't know anything about the order of $H_1 \cap H_2$..

You could explain here if i'm just missing somethign

Take $h_1 \in H_1 \cap H_2$, for any $g \in H_1$, $gh_1g^{-1} \in H_1$

@Skullpatrol Input from mathematicians is certainly very important, but mathematicians (a) are far from unanimous on how it should be taught, and (b) don’t always take into account practical constraints. I don’t think that they should be the only ones to provide input.

And, $l \in H_2$ $lh_1l^{-1} \in H_2$

@Kannappan: That's just the definition of normality

Yes, that is true because subgroups satisfy closure!

@JonasTeuwen I will have to think on that and give you a list. I can't think of just one that stands out.

@Kannappan: By this, any subgroup is normal..

@mk in its normalizer!

@BrianMScott I think a lot of the ambiguity is past down from generation to generation and the practical constraints should be left to the teaching psychologists.

You need $g^{-1}hg \in H_1 \cap H_2$ for each $g in H_1$, $h in H_1 \cap H_2$

not g^{-1}hg \in H_1..

@Skullpatrol In all honesty, I’m a bit skeptical of them, too!

@BrianMScott Me too.

@mk I seem to have made some obvious mistakes! I need to think a bit now!

@Kannappan: Sorry, I just don't think this is right at all

@BrianMScott But the meat of the subject should be left to be presented as the originators want it to be, in my opinion.

How do you get latex visible in here?

@mk But please do not down vote! I'll edit it now =)

@mk With @robjohn, he best answers this question!

@mk Thank you, I was about to ask that too. Reading Latex code is painful.

@Skullpatrol I certainly agree that the mathematics itself shouldn’t be misrepresented, but a good teacher will adjust the actual presentation to the students. The real problem is that schoolteachers can only teach what they themselves understand, and any guidelines have to take that into account.

robjohn does group theory?

@anon: I think he meant the thing about latex

@mk Take a look at this.

@fortuon: Okay, found it math.ucla.edu/~robjohn/math/mathjax.html

That’ll work too.

This is magic!

@BrianMScott Where is this "clip board?"

On Windows XP

It’s the internal Windows clipboard that you use when you cut and paste. Just paste it into the location field of a bookmark. In other words, instead of bookmarking a URL, you’ve bookmarked that bit of JavaScript.

ctrl + V

The clipboard refers to whatever you've copied and have for pasting.

Ohh... I get it now thanks guys

;-)

Just remember that if you refresh the page, you’ll have to refresh the bookmarklet as well.

Got it.

Thanks again.

@m.k. Looks like the proof is irreparable!

I have fallen a trap to thinking in the air!

Air today and gone tomorrow. sigh

@Kannappan: That happens sometimes

@BrianMScott what does that mean? ;(

It’s a pun on the somewhat traditional ‘Here today and gone tomorrow’, referring to something short-lived. It was perhaps also an oblique reference to ‘building castles in the air’, meaning building something on a very infirm foundation.

It was your thinking in the air that triggered it.

Alright! I need to think about a rectification!

There's no hope...

@unNaturhal of what?

@m.k. I realise I should have claimed the product and not union!

@KannappanSampath, Have you ever seen Jimi Hendrix when he burns his Guitar? I'm doing tha same thing with my copybook D:

@robjohn If something pops up in your mind do tell me :-).

@MattN The answer will not, the books might.

@Dylan Can you do me a favour?

Can I ask you?

@Kannappan: What do you mean?

Let me think! I seem to have got a proof! Nothing about those products and blahs!

@mk I seem to filled one half of the gap. I'll just post that and the other I'll still think!

OK, now I'm really irritated... there's a paper with a proof I need, but my library doesn't seem to have an online subscription to the journal...

@ZhenLin May be, I can check out if my library does subscribe?

This one, but I've just noticed it's part of a book... which is probably why we're not subscribed

@KannappanSampath What's up?

@DylanMoreland This problem has been evading me for quite some time! Can we think about thta?

@Jonas: I still like your answer, but I don't see why the other answers have gone without votes (I have upvoted them both just now).

Let me read it.

@ZhenLin what paper/proof? never mind

@ZhenLin We have a subscription to Springerlink but the institute network is not logged in right now (here, 5.14 am) . May be tomm. it is possible, I think I am not sure!

@ZhenLin, I can access the paper. How do you want me to send/show it to you?

That would be much appreciated.

But how?

Email, upload site, xdcc, ....

@anon FWIW, I'd prefer first mode

Just post a link here and delete it quickly, or something...

lol

Thanks!

Tell me when you're done.

@Dylan Are you still around!

Anybody interested in writing something to generate some numbers for any of my questions over here for research purposes? I haven't wrote code in years and the only language I know is basic, too tedious to do by hand, etc. Also, who thinks my comments would be better as a standalone question? Or: does anyone know any study/answers to them?

@anon I am out of the game!

With any luck I imagine there could be something of interest.