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# User talk:Arcan

 World Blogs Clogs Elections The Bar (talk) (talk) (talk) (talk) (hic)
Welcome to RationalWiki, Arcan!

Check out our guide for newcomers and our community standards!

Tell us how you found RationalWiki here!

If you are interested in contributing:

Are you a sock of Cracker, Human, or Bohdan?--PalMD-Did that sound a little harsh? 22:47, 5 February 2008 (EST)

No, I'm not, and no, it didn't. :-) -Arcan 00:43, 6 February 2008 (EST)

## Teacher

Your a teacher? I have genuine respect for you now. -- Radioactive afikomen Please ignore all my awful pre-2014 comments. 03:09, 6 March 2008 (EST)

Haha, well, let's hold off until I actually do something to earn that respect. Also, I'm not a teacher yet. One has to be employed to hold that title. While we're at it, and it might just be me being dense to humor, but you do realize what you did wrong there, yes? Arcan 03:41, 6 March 2008 (EST)
No, I think I did do something wrong. I am but an amateur grammar Nazi. : ) -- Radioactive afikomen Please ignore all my awful pre-2014 comments. 12:52, 6 March 2008 (EST)

## Not petty at all!

That's what I get for writing quotes by hand instead of copypasting them! Thanks for fixing :) --Sid 14:47, 23 March 2008 (EDT)

Haha, no problem. I really liked the entry, and just thought the poor grammar to be an absurd cherry on the absurd sunday. Arcan 14:49, 23 March 2008 (EDT)

## sig test

just testing --Arcan ¡ollǝɥ 13:03, 26 March 2008 (EDT)

ok, trying again... --Arcan ¡ollǝɥ 13:06, 26 March 2008 (EDT) Yay!

## Coverstory issues

We are trying to make the inclusions teh more awesome. Sadly, we have made teh misteaks along the way. Please to kindly hang on while we unmeak our errors and meak things perfect. I'd link to where we are working on this, but I forgot the darn page! Probably one of the "coverstory" talk pages, but who knows? In the meantime, thanks for taking out the trash. human 01:20, 16 April 2008 (EDT)

Ok, I don't think I understood any of that... --Arcan ¡ollǝɥ 01:22, 16 April 2008 (EDT)
Alright, just saw the talk page for the Behe article, so I'm guessing this is in reference to that, but I think I'm still confused. Are you saying it was good that I commented? Bad? I didn't mean to offend, and I hope it wasn't taken that way. --Arcan ¡ollǝɥ 01:25, 16 April 2008 (EDT)
What you did was fine. The "noinclude" was a so-far-failing attempt to specify more precisely in each cover story what gets transcluded to the main page. human 13:57, 16 April 2008 (EDT)
Ah, ok, good. I thought I'd screwed something up or made some faux pas. --Arcan ¡ollǝɥ 13:59, 16 April 2008 (EDT)
Oh wait! I was just thinking - the Behe article didn't have the noinclude. Could that have been the problem? --Arcan ¡ollǝɥ 14:01, 16 April 2008 (EDT)

## English Teacher

Just saw you say that you are an English teacher. A real one or TEFL like me?--Bobbing up 10:54, 24 May 2008 (EDT)

I'm not sure what TEFL means (I'm guessing English as a Foreign Language, but I'm not sure). I'm actually a certified secondary (67th-12th) English teacher. However, this is a recent development (as of the beginning of this year) so I'm still unemployed, but a teacher nonetheless. --Arcan ¡ollǝɥ 11:12, 24 May 2008 (EDT)
Yes, TEFL is Teaching English as a Foreign Language. Most expatriate English teachers have some form of TEFL qualification - or should have one. Good luck looking for work. :-) --Bobbing up 11:17, 24 May 2008 (EDT)

 This user is a Grammar Nazi. Heil Stickler!

Wazza (Not Wazzock, Wazza)Approach the Presence 11:22, 24 May 2008 (EDT)

Haha, so you've not ready any of my contribs, have you? I'm very much a "descriptivist." I only get uppity about grammar etc. in formal situations - and I made that very clear to my students as well. It's kinda asinine to work about SWE in informal communication. Oh, and I cuss like a sailor. I don't even realize when I'm doing it (though I filter it the classroom). I'm very much not the typical English teacher. --Arcan ¡ollǝɥ 11:29, 24 May 2008 (EDT)
Consider a career in TELF, you sound like a perfect match for the ne'er-do-wells, alcoholics, and various types of escapee who seem to end up in teh profession overseas. :-) --Bobbing up 11:36, 24 May 2008 (EDT)
I'm going to take that as a compliment, whether you meant it that way or not ;-) Honestly though, I love teaching kids, and the "rogue" element of my style really seems to work well. It's so great to basically watch their minds form (at the 7th grade level) and then serve as a guide when their minds are kinda taking their first steps (at the upper grades). To help in that process and encourage it and to know that you're a part of it, it is literally awesome. I'm not ready to give up the traditional classroom yet. I was thinking of doing this Japanese program that my friend applied to, but I didn't think I was quite ready for that. Eventually, maybe :-) --Arcan ¡ollǝɥ 11:52, 24 May 2008 (EDT)
Good luck with that then. :-) I used to like kids - till I had to teach them. On the other hand the difficult sods are aged 13+ - before that age they they all want to be your friend, which is sort of sweet.--Bobbing up 11:58, 24 May 2008 (EDT)
Yeah, that's about the age that I get them, 12-13 and up. And some of them are monsters, most definitely. I guess I'm just that crazy :-) --Arcan ¡ollǝɥ 12:01, 24 May 2008 (EDT)
I guess that everybody is different. Now I only teach adults.--Bobbing up 13:29, 24 May 2008 (EDT)

## an EXTREMELY awkward question

Is it only coincidence that your screen name is the almost same as that of Arkan, the notorious mass-murderer and leader of a nasty Serbian paramilitary group during the Bosnian war? PFoster 12:41, 10 June 2008 (EDT)

Wow, yeah, the awkardomer is off the charts. Considering I'd never heard of Arkan, yes, it is a total coincidence. "Arcan" is actually an abbreviaion. Sometimes I go by "ArcanRed," which is two abbreviations. Abbreviations of what, though, is a mystery I keep to myself. So no, I'm not glorifying any mass-murderers, etc. It's mainly just a nonsense word. But thanks for asking, I think... --Arcan ¡ollǝɥ 14:07, 10 June 2008 (EDT)
Hey Arcan - sorry if the question freaked you out - it's just that I did a lot of work on Serb nationalism and the breakup of Jugoslavia in my undergrad, so I had to ask. I actually figured it would make no sense for a Serb apologist to be posting here, but I had to ask. PeacePFoster 21:37, 10 June 2008 (EDT)
No problem. I didn't really freak me out so much as it was totally out of left field for me. I'd never heard of Arkan. You just gave me a bit of homework. ;-) --Arcan ¡ollǝɥ 10:09, 11 June 2008 (EDT)

## Thanks!

Thanks for going through all those links! Only Crocodite could do something that slimey. I really hope PJR turns up and restores the history, or else we've lost all our lulz :( <font=""; face="Comic Sans MS">Jellyfish!I kill spider 22:23, 11 June 2008 (EDT)

If it were just to burn the evidence, I think think PJR might come to our rescue (indirectly, of course), but the "vulgar link" will likely keep PJR away. Most of the best WIGOs (10+) didn't link to the Mainpage Talk, so they're in the clear. The only 10+ one I couldn't fix was "Words can't express the stupid." There's just far too much stupid to know what the reference was to. Thanks for the thanks. ;-) --Arcan ¡ollǝɥ 22:31, 11 June 2008 (EDT)
Thanks for the thanks for the thanks! We could be here some time. <font=""; face="Comic Sans MS">Jellyfish!I kill spider 22:34, 11 June 2008 (EDT)
That was the one that goes more or less like this:
Our heroes are in the middle of a debate (about gun control, I think), after a barrage of useless comments, one contender tries to put some organization.
• Some Guy: I've made this new section for people in the debate to show your empirical evidence that supports your case.
• Andy: Why evidence? Surely pure logic is enough.
NightFlareStill doesn't have a RWW article. 23:43, 11 June 2008 (EDT)
Yes you do, whiner ;) ħuman 02:38, 12 June 2008 (EDT)
I think I remember that one, though the Mainpage Talk archives all start to sound alike after a while. I'll let you add that one, if you like. I know, I know, I'm so generous. ;-) Oh, and Jellyfish, if you're still watching, thanks for the thanks for the thanks. Of course, I demand last wordism, being the filthy liberal that I am. --Arcan ¡ollǝɥ 00:04, 12 June 2008 (EDT)
I'm always watching, Archy, and I demand that you cease your lastwordism. This isn't Wikipedia! We don't allow liberals to say what they like. Therefore -- thanks for the thanks for the thanks for the thanks.
Kowardjou has been instructed to firebomb you if you try to reply :) <font=""; face="Comic Sans MS">Jellyfish!I kill spider 00:11, 12 June 2008 (EDT)
Right, like that could ever- OH MY GOD WHAT'S THAT IN THE SKY! NOOOO! --Arcan ¡ollǝɥ 01:28, 12 June 2008 (EDT)
That's Crockospite, actually. He has way too much time on his hands. <font=""; face="Comic Sans MS">Jellyfish!I kill spider 01:35, 12 June 2008 (EDT)

## Three Pointzes

This user has officially earned THREE POINTZES for correctly identifying the key in the challenge at talk:WIGO CP. Congratulations, and pleez to spend them not awl in teh saim plaice. And pleez also to yuze them for gud and not ebil. ħuman 23:42, 14 June 2008 (EDT)

Let's see, do you want the glow-in-the-dark fake vampire teeth, or the koosh ball key-chain? Radioactive afikomen Please ignore all my awful pre-2014 comments. 00:07, 15 June 2008 (EDT)
No way, I'm sooo saving up for the stuff on the wall! --Arcan ¡ollǝɥ 00:17, 15 June 2008 (EDT)
Good plan, Arcan. The bigger prizes are so worth saving up for. Now it's time for the rigged basketball toss... ħuman 00:40, 15 June 2008 (EDT)

## Seven or Eight Pointzes!!!

For being a Lert in teh middlez of the night and cranking back on wandalism. Thanks, Arcan. You were perfect. ħuman 03:19, 28 June 2008 (EDT)

## Public education

I've just kicked off the article public education (which I'm surprised we already didn't have). I feel it has the potential to become something truly great, as it touches upon so many aspects of our mission (which may not be readily apparent, given its magnificently cruddy form at the moment). Being a public school teacher, I would greatly appreciate any input you have to offer. Radioactive afikomen Please ignore all my awful pre-2014 comments. 06:34, 2 July 2008 (EDT)

Are you saying my desperate begging for help is usually lacking? Radioactive afikomen Please ignore all my awful pre-2014 comments. 06:39, 2 July 2008 (EDT)
It's usully more along the lines of "BAAAWWWWW, I phail, plz hlp me!" (Paraphrasing slightly for humorous effect, of course.) --AKjeldsenCum dissensie 06:44, 2 July 2008 (EDT)
I'll help out however I can. :-) --Arcan ¡ollǝɥ 21:03, 2 July 2008 (EDT)
Ah, thank you. Radioactive afikomen Please ignore all my awful pre-2014 comments. 22:39, 2 July 2008 (EDT)

## You = sysop

Thanks! Wow, givin' 'em out like candy, haha. Well, I solemnly pledge that this will probably not change anything, considering I have no idea what all these new buttons etc. do. I mean, I can kinda guess what "lock" and "vaporise" are, but I think I'll be too worried about causing some sort of catastrophic system failure to use them. Still, thanks! --Arcan ¡ollǝɥ 15:01, 24 July 2008 (EDT)
Thanks for the thanks. (I love doing this)
No need to worry, Arcan, I more or less set the lowest possible standard for the current batch of demotions already. By comparison, your ability to cause problems should be minimal. --AKjeldsenCum dissensie 15:26, 24 July 2008 (EDT)

## RWV

Good idea. ~ Gloom(is never asleep) 18:12, 26 July 2008 (EDT)

I've got it as a toolbar button.   ContribsTalk 18:14, 26 July 2008 (EDT)
Really? Where? ~ Gloom(is never asleep) 18:16, 26 July 2008 (EDT)
(points→) Here.   ContribsTalk 18:19, 26 July 2008 (EDT)
Sigh...
Screenshot? ~ Gloom(is never asleep) 18:21, 26 July 2008 (EDT)
I'll wager a guess that she put it in the browser toolbar. --AKjeldsenCum dissensie 18:29, 26 July 2008 (EDT)
You'd win. SS coming up - don't hold yer bref - (don't like GIMP enuff yet)   ContribsTalk 18:35, 26 July 2008 (EDT)
Image:SS02.gif There.   ContribsTalk 18:37, 26 July 2008 (EDT)
Nice. I thought you meant it was on the navigation panel in RW somehow. I apoly guise.~ Gloom(is never asleep) 18:40, 26 July 2008 (EDT)
Could be put there.   ContribsTalk 18:42, 26 July 2008 (EDT)

## Diablo

So like "block" would be a frost mage skill right? Ready for Diablo 3? tmtoulouse irritate 23:20, 25 August 2008 (EDT)

I will be ready for it, when it eventually comes out (and I even like the art style). Blizzard has so many other things in the queue, though. I'm not holding my breath for it. --Arcan ¡ollǝɥ 23:22, 25 August 2008 (EDT)
Like what? Wraith of the Lich King? Who cares about that? Really? So your not one of the people crying that its too colorful and not "gothic" enough? tmtoulouse irritate 23:25, 25 August 2008 (EDT)
They've also got Starcraft 2 coming up, and while these may or may not be games I'm looking forward to, I doubt Blizzard is going to release them all that close to one another. And yes, I do not support the Necromartyr and his ilk. --Arcan ¡ollǝɥ 23:30, 25 August 2008 (EDT)
Oh yea Starcraft 2.......forgot about that........ tmtoulouse irritate 23:41, 25 August 2008 (EDT)
Yeah, and they won't want to step on those toes. --Arcan ¡ollǝɥ 23:43, 25 August 2008 (EDT)
Meh, I like most of what they put out...then again I like most computer games. What I am really excited about though is Spore. tmtoulouse irritate 23:48, 25 August 2008 (EDT)
I've never really gotten into pure strategy games, be they TBS or RTS, though civ games (but not Civ proper) do draw me in. Blizzardwise, I did get sucked into Diablo II and WoW for a while each. I'm tentatively excited for Spore - I'm worried that the parts won't connect properly (e.g. what you do in the single-cell stage won't carry over much into the tribe stage, and so on), and that's only exasperated by the fact that you can jump in at any stage. Still, though, it's Will Wright, and he tends to deliver on his promises. If it were Peter Molyneux, then I'd be concerned.
I haven't followed the details enough, I enjoyed the "creature creator" that came out and have seen vids and what not. I am looking forward to it. Since I um procure most my video games from torrents my standards are lower than if I paid \$50. tmtoulouse irritate 00:02, 26 August 2008 (EDT)
Haha, I understand. I've been trying to get away from the pirate life (very recently, as in after Assassin's Creed), which means that I mostly play free/cheap games and only dole out money when I can be sure I'll enjoy it, like the Orange Box, Mass Effect, and most recently, Beyond Good and Evil (from Steam). --Arcan ¡ollǝɥ 00:06, 26 August 2008 (EDT)
I justify my pirate life by virtue that if I didn't download it I wouldn't buy it. My graduate student stipend doesn't support gaming, or movies for that matter. Yea thats my justification. So Mass Effect any good? I am 75% of my way there...... tmtoulouse irritate 00:41, 26 August 2008 (EDT)
I definitely understand - "just-out-of-college-and-still-unemployed" doesn't support much either. As for Mass Effect, I'm nearly finished it, and I'm actually sad about that, because I'd like to keep playing it. I absolutely loved it. However, the "Citadel" section (near the beginning of the game) is dull as hell. Very little story or exposition, no action whatsoever, not-quite-even-fetch-quest quests, etc. Once past that, the game pick up fantastically. The driving sections are rubbish until you get used to them, but I really enjoyed the combat and "diplomacy" bits. Then again, KOTOR is pretty much my favorite game ever, and I liked Jade Empire as well, so take that as you will. --Arcan ¡ollǝɥ 00:49, 26 August 2008 (EDT)
Any game I really support I will usually buy, I have even gone out and purchased games I have pirated and really liked. I see it as a form of voting. I will probably buy Spore, as well as Diablo III (I am a sucker for the hack and slash, being able to play a game only by pointing and clicking really revs up those dopamine cells). I might even take a look at the Warcraft expansion but that is dangerous, I dumped more time than I could afford when I played. Anyway, I have been lately messing with Titan's Quest while I wait for Mass Effect to download. I remember Beyond Good and Evil but I never went far with it. I have Overlord, S.t.a.l.k.e.r, and a few others all sitting in the sorta liked but didnt really catch on category. I am just rambling I will stop :). tmtoulouse irritate 00:56, 26 August 2008 (EDT)

<- I'll probably buy Spore, and I certainly hope to have a job/money by the time Diablo III comes out. I haven't played Titan's Quest. For some reason it was totally off my radar when it first came out, and every time I remember it I'm in the middle of something else. BG&E is another of my favorites, but I like platformers. Still, it's wholly unique in what it does. I actually finished Overlord; it's fun, but oddly unpolished at times. It's a lot like a cross between Pikmin and Fable, whose sequel I'm also excited for (even though Molyneux always overpromises), with voice actors lifted directly from the latter. S.t.a.l.k.e.r. was very good, but overall I think I respected and appreciated it more than I liked it, if that made any sense. I'm sure I'll try the sequel eventually (even though I might have to return to my old ways to try it). Ok, now I'm rambling. Haha. --Arcan ¡ollǝɥ 08:56, 26 August 2008 (EDT)

## Goatspeed!

Thanks for that reversion to my user page. Seriously, thank you. =) ĴαʊΆʃÇä₰ No comment

Haha, you would have gotten it soon. I just refreshed at the right time. :-) --Arcan ¡ollǝɥ 23:19, 15 September 2008 (EDT)
Thanks for fixing my user page.--Bobbing up 13:58, 20 October 2008 (EDT)
No problem :-) --Arcan ¡ollǝɥ 14:32, 20 October 2008 (EDT)
Ditto! Thanks. --PsyGremlinWhut? 17:02, 20 October 2008 (EDT)

## Sorry...

I accidentally reverted your reversion of a vandal's work. I just reverted my reversion of your reversion, but I thought you might want to know. (I still don't quite know how to use wikis very well...) A Writer of Vaudevilles 22:50, 18 November 2008 (EST)

Haha, I saw that, but I didn't think anything of it. The nice thing about wikis is that nearly anything that can be broken can be easily unbroken, so don't worry about it :-) --Arcan ¡ollǝɥ 22:58, 18 November 2008 (EST)

## Barnstar

[Award removed]

Thanks for the thought, but any award given to Andrew Usher Fall Over isn't one I'd like to get. --Arcan ¡ollǝɥ 20:58, 12 December 2008 (EST)

## Thanks

I was thinking about removing that from the WIGO as well, but was a bit too hesitant, I guess. --Kels 18:13, 25 December 2008 (EST)

Haha, no problem :-) --Arcan ¡ollǝɥ 18:29, 25 December 2008 (EST)

## Thanks 2

Cheers for cleaning up all that page move vandalism. - User ${\displaystyle \scriptstyle {\sqrt {12}}\sum _{n=0}^{\infty }(-3)^{-n}(2n+1)^{-1}}$ 23:25, 2 January 2009 (EST)

Haha, no problem --Arcan ¡ollǝɥ 23:28, 2 January 2009 (EST)

## Teaching granny to suck eggs?

If you open the "move" log you can just click "revert" & move the associated "talk" by clicking the box. Toast 23:28, 2 January 2009 (EST)

Four things: 1. Huh? 2. Ah, I didn't know about the move log thing 3. I knew about the check box, but somehow, I unchecked it for the last move 4. Thanks, I think :-) --Arcan ¡ollǝɥ 23:31, 2 January 2009 (EST)

## Thanks 3

Ah thanks for correcting my WIGO, I was wondering what the problem was. --GTac 11:06, 20 January 2009 (EST)

Haha, no problem. Honestly, after I copy/pasted the edit comment, I was worried it would look like I was shouting, instead of just quoting. Glad to help, and glad to hear no offense was taken! --Arcan ¡ollǝɥ 11:17, 20 January 2009 (EST)

## Rolled you back in the Saloon...

...sorry, my bad, was trying to fix CUR's accidental removal of a comment...TheoryOfPractice 22:06, 21 January 2009 (EST)

Wasn't removing a comment. I was trying to get my paragraph after an edit conflict. God Darwin, I hate those things! --"ConservapediaUndergroundInductorrecent changes patroller 22:09, 21 January 2009 (EST)

I SAID "accidental..."TheoryOfPractice 22:11, 21 January 2009 (EST)

No problem, I figured out what was going on and just fixed it :-) --Arcan ¡ollǝɥ 22:10, 21 January 2009 (EST)

## Extra WIGO warning

Yeah, it's probably not necessary. I knew it was a matter of time before I got caught with my IPants down.

## thanks!

Thanks for improving my part when trying to post the latest regarding to an MSNBC poll in Conservapedia:What is going on at CP? I'm going to "infiltrate" into Conservapedia by creating a fake account to expose their extreme homophobia. I remembered about a year ago, I tried to question their homophobia in their site. Obviously, they banned me for it. Now for some nice revenge, read more here. --Dark Paladin X 13:24, 22 April 2009 (EDT)

I'm not sure if you're watching this page, but I'd advise against your plan. We already know they're homophobic. Ken's Summa Homosexualita proves that. I'm not sure what you "revenge" would be, and normally I'm all for justice and/or vengeance, but there's no slight to avenge here. They blocked you from a place that would only cause you stress. Don't stoop to their level. Be better than they are. Living well is the best revenge. --Arcan ¡ollǝɥ 15:04, 22 April 2009 (EDT)

## Wigo "Seemingly"

Weasel wordsn- Nice. Don't let the truth spoil a good story. Tricksy 09:20, 21 May 2009 (UTC)

Your intentions and your actions didn't line up there. Whatever you meant to say, to everyone else it looked like you called him a troll. So, as someone not personally involved in that WIGO, "seemingly" is entirely accurate in that case. It reports what you said (not to mention links to exactly what you said) without claiming to know what you intended, but doesn't lose the interpretation that most people looking at your post would glean. Even the person you were talking about misunderstood you. There is an update in the same WIGO about your clarification, so your side of the story is told. As it is, you're trying to revise a story after it happened about which you literally cannot be objective, because you were not just an active participant, but the focus of it.
BTW - Thanks for the accusation. Maybe if you didn't imply I was a liar, I'd be less hostile. And I really appreciate you making this personal, rather than bringing it up on the talk page. Classy. --Arcan ¡ollǝɥ 19:05, 21 May 2009 (UTC)
"Seemingly": fair enough I suppose. That's why I re-cast it, but "seemingly" is an oft-use weasel word employed to make statements that would otherwise be taken as dishonest. I accept that this is not what you were doing. In my post above, I used a cliche to sum up my perception of WIGO aSK lately. I apologise for any accusation that you were being dishonest. I stress though, that I absolutely was talking about Gulik's post and not Gulik personally. As to Gulik's misunderstanding, I have already addressed that at aSK. Actually the focus of the WIGO was Gulik's weird and humourous implication that photos taken by YEC's aren't reliable, but someone else revised the story. Apparently as a participant in the thread, I should not have WIGOed it, but followed the accepted practice of posting it into talk to prompt someone else to WIGO it for me. Enough, I am rambling. Again, I apologise for any accusations against you personally. Tricksy 01:25, 22 May 2009 (UTC)

## 'Troll'

I'm afraid I don't live underneath a bridge, but I highly suspect you live in your parents basement. What does that make you? MarcusCicero 21:59, 10 June 2009 (UTC)

You really aren't very good at this. --Arcan ¡ollǝɥ 03:02, 11 June 2009 (UTC)
Yeah that is pretty lame. MC reminds me of jinx. Ace McWickedDisco Jesus 04:26, 11 June 2009 (UTC)
Not wrong then ;) MarcusCicero 08:46, 11 June 2009 (UTC)
You used to be better than this. There was a time when an MC post would have this place in upheaval for days! But now, this isn't even CUR-level trolling. Seriously now, did you expect me to be all insulted and scan a copy of my lease? Or perhaps go with the "I'm rubber and you're glue" defense, similar to your comment at the start of this section? Where has that magic gone, MC? I mean, I didn't like you then, but at least you used to be interesting. Now, you're just kinda sad. It's almost pitiable. --Arcan ¡ollǝɥ 09:09, 11 June 2009 (UTC)
Its a natural progression. I don't come on here solely to antagonise. Especially at the start I actually cared about the place. I'm not a troll. 86.45.200.112 12:28, 11 June 2009 (UTC)

## Human WIGO

I reinstated the WIGO but commented it out. It already had a vote (from the poster I presume) and any new WIGO wouldn't know that this number had already been used. ГенгисRationalWiki GOLD member 21:39, 22 July 2009 (UTC)

I thought about commenting it out, but there was only one vote, so I downvoted it to zero it out. I guess it didn't work, though. Thanks! --Arcan ¡ollǝɥ 01:39, 23 July 2009 (UTC)

Is that you?

## WIGO

It is getting voted down because it is the third time it has been up there now. - π 14:07, 25 September 2009 (UTC)

Okay it does appear to be an update of the previous one, maybe it would be better to present it as such. - π 14:11, 25 September 2009 (UTC)
I didn't write the first one, but if it's a repeat, maybe it should be commented out? I honestly didn't think the whole event was that funny, but I thought I could at least maybe improve the WIGO. Is it even worth updating the previous one? --Arcan ¡ollǝɥ 19:33, 25 September 2009 (UTC)
Yes it is an update. From August, look for "Andy proves his complete illiteracy", #2343. AP 02:54, 26 September 2009 (UTC)

## Sorry

Nice catch on that one, guess I wasn't reading as carefully as I should. --Opcn (talk) 06:40, 31 March 2010 (UTC)

Don't worry about it. Sorry if I came off a little harsh! --Arcan ¡ollǝɥ 06:44, 31 March 2010 (UTC)


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# Power law

(Redirected from Power-law)
An example power-law graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule).

In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.[1]

## Empirical examples

The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares,[2] the foraging pattern of various species,[3] the sizes of activity patterns of neuronal populations,[4] the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms,[5] the sizes of power outages, criminal charges per convict, volcanic eruptions,[6] human judgements of stimulus intensity[7][8] and many other quantities.[9] Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.

## Properties

### Scale invariance

One attribute of power laws is their scale invariance. Given a relation ${\displaystyle f(x)=ax^{-k}}$, scaling the argument ${\displaystyle x}$ by a constant factor ${\displaystyle c}$ causes only a proportionate scaling of the function itself. That is,

${\displaystyle f(cx)=a(cx)^{-k}=c^{-k}f(x)\propto f(x),\!}$

where ${\displaystyle \propto }$ denotes direct proportionality. That is, scaling by a constant ${\displaystyle c}$ simply multiplies the original power-law relation by the constant ${\displaystyle c^{-k}}$. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both ${\displaystyle f(x)}$ and ${\displaystyle x}$, and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws (e.g., if the generating process of some data follows a Log-normal distribution).[citation needed] Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below.

### Lack of well-defined average value

A power-law ${\displaystyle x^{-k}}$ has a well-defined mean over ${\displaystyle x\in [1,\infty )}$ only if ${\displaystyle k>2}$, and it has a finite variance only if ${\displaystyle k>3}$; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior.[10] This can be seen in the following thought experiment:[11] imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US\$. What happens to the average income in the room? Income is distributed according to a power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2).

On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis).[citation needed] On the other hand, this also allows for cost-efficient interventions.[11] For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially.[12]

The median does exist, however: for a power law xk, with exponent ${\displaystyle k>1}$, it takes the value 21/(k – 1)xmin, where xmin is the minimum value for which the power law holds.[13]

### Universality

The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of the system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality—can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO2 at their boiling points fall in the same universality class because they have identical critical exponents.[citation needed][clarification needed] In fact, almost all material phase transitions are described by a small set of universality classes. Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor. Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

## Power-law functions

Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them.[14] The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems;[15] see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, neuroscience, sociology, economics and more.

However, much of the recent interest in power laws comes from the study of probability distributions: The distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory), which considers the frequency of extremely rare events like stock market crashes and large natural disasters. It is primarily in the study of statistical distributions that the name "power law" is used.

In empirical contexts, an approximation to a power-law ${\displaystyle o(x^{k})}$ often includes a deviation term ${\displaystyle \varepsilon }$, which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from the power-law function (perhaps for stochastic reasons):

${\displaystyle y=ax^{k}+\varepsilon .\!}$

Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: ${\displaystyle p(x)=Cx^{-\alpha }}$ for ${\displaystyle x>x_{\text{min}}}$ where the exponent ${\displaystyle \alpha }$ (Greek letter alpha, not to be confused with scaling factor ${\displaystyle a}$ used above) is greater than 1 (otherwise the tail has infinite area), the minimum value ${\displaystyle x_{\text{min}}}$ is needed otherwise the distribution has infinite area as x approaches 0, and the constant C is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see power-law probability distributions below for details. Typically the exponent falls in the range ${\displaystyle 2<\alpha <3}$, though not always.[9]

### Examples

More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income).[16] Among them are:

#### Criminology

• number of charges per criminal offender

#### Biology

• Kleiber's law relating animal metabolism to size, and allometric laws in general
• The two-thirds power law, relating speed to curvature in the human motor system.
• The Taylor's law relating mean population size and variance of populations sizes in ecology
• Neuronal avalanches[4]
• The species richness (number of species) in clades of freshwater fishes[18]
• The Harlow Knapp effect, where a subset of the kinases found in the human body compose a majority of published research[19]

#### Meteorology

• The size of rain-shower cells,[20] energy dissipation in cyclones,[21] and the diameters of dust devils on Earth and Mars [22]

#### Economics

• Population sizes of cities in a region or urban network, Zipf's law.
• Distribution of artists by the average price of their artworks.[29]
• Distribution of income in a market economy.
• Distribution of degrees in banking networks.

### Variants

#### Broken power law

Some models of the initial mass function use a broken power law; here Kroupa (2001) in red.

A broken power law is a piecewise function, consisting of two or more power laws, combined with a threshold. For example, with two power laws:[30]

${\displaystyle f(x)\propto x^{\alpha _{1}}}$ for ${\displaystyle x
${\displaystyle f(x)\propto x_{\text{th}}^{\alpha _{1}-\alpha _{2}}x^{\alpha _{2}}{\text{ for }}x>x_{\text{th}}}$.

#### Power law with exponential cutoff

A power law with an exponential cutoff is simply a power law multiplied by an exponential function:[31]

${\displaystyle f(x)\propto x^{\alpha }e^{\beta x}.}$

#### Curved power law

${\displaystyle f(x)\propto x^{\alpha +\beta x}}$[32]

## Power-law probability distributions

In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of ${\displaystyle x}$,[33]

${\displaystyle P(X>x)\sim L(x)x^{-(\alpha -1)}}$

where ${\displaystyle \alpha >1}$, and ${\displaystyle L(x)}$ is a slowly varying function, which is any function that satisfies ${\displaystyle \lim _{x\rightarrow \infty }L(r\,x)/L(x)=1}$ for any positive factor ${\displaystyle r}$. This property of ${\displaystyle L(x)}$ follows directly from the requirement that ${\displaystyle p(x)}$ be asymptotically scale invariant; thus, the form of ${\displaystyle L(x)}$ only controls the shape and finite extent of the lower tail. For instance, if ${\displaystyle L(x)}$ is the constant function, then we have a power law that holds for all values of ${\displaystyle x}$. In many cases, it is convenient to assume a lower bound ${\displaystyle x_{\mathrm {min} }}$ from which the law holds. Combining these two cases, and where ${\displaystyle x}$ is a continuous variable, the power law has the form

${\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha },}$

where the pre-factor to ${\displaystyle {\frac {\alpha -1}{x_{\min }}}}$ is the normalizing constant. We can now consider several properties of this distribution. For instance, its moments are given by

${\displaystyle \langle x^{m}\rangle =\int _{x_{\min }}^{\infty }x^{m}p(x)\,\mathrm {d} x={\frac {\alpha -1}{\alpha -1-m}}x_{\min }^{m}}$

which is only well defined for ${\displaystyle m<\alpha -1}$. That is, all moments ${\displaystyle m\geq \alpha -1}$ diverge: when ${\displaystyle \alpha \leq 2}$, the average and all higher-order moments are infinite; when ${\displaystyle 2<\alpha <3}$, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge – as more data is accumulated, they continue to grow. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails.

A modification, which does not satisfy the general form above, with an exponential cutoff,[9] is

${\displaystyle p(x)\propto L(x)x^{-\alpha }\mathrm {e} ^{-\lambda x}.}$

In this distribution, the exponential decay term ${\displaystyle \mathrm {e} ^{-\lambda x}}$ eventually overwhelms the power-law behavior at very large values of ${\displaystyle x}$. This distribution does not scale and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. (Note that the pure form above is a subset of this family, with ${\displaystyle \lambda =0}$.) This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects.

The Tweedie distributions are a family of statistical models characterized by closure under additive and reproductive convolution as well as under scale transformation. Consequently, these models all express a power-law relationship between the variance and the mean. These models have a fundamental role as foci of mathematical convergence similar to the role that the normal distribution has as a focus in the central limit theorem. This convergence effect explains why the variance-to-mean power law manifests so widely in natural processes, as with Taylor's law in ecology and with fluctuation scaling[34] in physics. It can also be shown that this variance-to-mean power law, when demonstrated by the method of expanding bins, implies the presence of 1/f noise and that 1/f noise can arise as a consequence of this Tweedie convergence effect.[35]

### Graphical methods for identification

Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots),[citation needed] mean residual life plots[36][37] and log–log plots. Another, more robust graphical method uses bundles of residual quantile functions.[38] (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data".

Pareto Q–Q plots compare the quantiles of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to the quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points " asymptotically converge" to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index ${\displaystyle \alpha }$ (also called Pareto index) is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails.[38]

On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i-th order statistic versus the i-th order statistic, for i = 1, ..., n, where n is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to "stabilize" about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots [39]

A straight line on a log–log plot is necessary but insufficient evidence for power-laws, the slope of the straight line corresponds to the power law exponent.

Log–log plots are an alternative way of graphically examining the tail of a distribution using a random sample. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot.[40][41] This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to "converge" to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published.[42] A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data.

Another graphical method for the identification of power-law probability distributions using random samples has been proposed.[38] This methodology consists of plotting a bundle for the log-transformed sample. Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions,[43][44][45][46][47][48][49] which provide a full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of ${\displaystyle \alpha }$, and do not demand the collection of much data).[citation needed] In addition, other types of tail behavior can be identified using bundle plots.

### Plotting power-law distributions

In general, power-law distributions are plotted on doubly logarithmic axes, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) cumulative distribution (cdf), ${\displaystyle P(x)=\mathrm {Pr} (X>x)}$,

${\displaystyle P(x)=\Pr(X>x)=C\int _{x}^{\infty }p(X)\,\mathrm {d} X={\frac {\alpha -1}{x_{\min }^{-\alpha +1}}}\int _{x}^{\infty }X^{-\alpha }\,\mathrm {d} X=\left({\frac {x}{x_{\min }}}\right)^{-\alpha +1}.}$

Note that the cdf is also a power-law function, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the ${\displaystyle n}$ observed values in ascending order, and plot them against the vector ${\displaystyle \left[1,{\frac {n-1}{n}},{\frac {n-2}{n}},\dots ,{\frac {1}{n}}\right]}$.

Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided.[50][51] The cdf, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a cdf representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.

### Estimating the exponent from empirical data

There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers. Some of the most reliable techniques are often based on the method of maximum likelihood. Alternative methods are often based on making a linear regression on either the log–log probability, the log–log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent.[9]

#### Maximum likelihood

For real-valued, independent and identically distributed data, we fit a power-law distribution of the form

${\displaystyle p(x)={\frac {\alpha -1}{x_{\min }}}\left({\frac {x}{x_{\min }}}\right)^{-\alpha }}$

to the data ${\displaystyle x\geq x_{\min }}$, where the coefficient ${\displaystyle {\frac {\alpha -1}{x_{\min }}}}$ is included to ensure that the distribution is normalized. Given a choice for ${\displaystyle x_{\min }}$, the log likelihood function becomes:

${\displaystyle {\mathcal {L}}(\alpha )=\log \prod _{i=1}^{n}{\frac {\alpha -1}{x_{\min }}}\left({\frac {x_{i}}{x_{\min }}}\right)^{-\alpha }}$

The maximum of this likelihood is found by differentiating with respect to parameter ${\displaystyle \alpha }$, setting the result equal to zero. Upon rearrangement, this yields the estimator equation:

${\displaystyle {\hat {\alpha }}=1+n\left[\sum _{i=1}^{n}\ln {\frac {x_{i}}{x_{\min }}}\right]^{-1}}$

where ${\displaystyle \{x_{i}\}}$ are the ${\displaystyle n}$ data points ${\displaystyle x_{i}\geq x_{\min }}$.[2][52] This estimator exhibits a small finite sample-size bias of order ${\displaystyle O(n^{-1})}$, which is small when n > 100. Further, the standard error of the estimate is ${\displaystyle \sigma ={\frac {{\hat {\alpha }}-1}{\sqrt {n}}}+O(n^{-1})}$. This estimator is equivalent to the popular[citation needed] Hill estimator from quantitative finance and extreme value theory.[citation needed]

For a set of n integer-valued data points ${\displaystyle \{x_{i}\}}$, again where each ${\displaystyle x_{i}\geq x_{\min }}$, the maximum likelihood exponent is the solution to the transcendental equation

${\displaystyle {\frac {\zeta '({\hat {\alpha }},x_{\min })}{\zeta ({\hat {\alpha }},x_{\min })}}=-{\frac {1}{n}}\sum _{i=1}^{n}\ln {\frac {x_{i}}{x_{\min }}}}$

where ${\displaystyle \zeta (\alpha ,x_{\mathrm {min} })}$ is the incomplete zeta function. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for ${\displaystyle {\hat {\alpha }}}$ are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.

Further, both of these estimators require the choice of ${\displaystyle x_{\min }}$. For functions with a non-trivial ${\displaystyle L(x)}$ function, choosing ${\displaystyle x_{\min }}$ too small produces a significant bias in ${\displaystyle {\hat {\alpha }}}$, while choosing it too large increases the uncertainty in ${\displaystyle {\hat {\alpha }}}$, and reduces the statistical power of our model. In general, the best choice of ${\displaystyle x_{\min }}$ depends strongly on the particular form of the lower tail, represented by ${\displaystyle L(x)}$ above.

More about these methods, and the conditions under which they can be used, can be found in .[9] Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions.

#### Kolmogorov–Smirnov estimation

Another method for the estimation of the power-law exponent, which does not assume independent and identically distributed (iid) data, uses the minimization of the Kolmogorov–Smirnov statistic, ${\displaystyle D}$, between the cumulative distribution functions of the data and the power law:

${\displaystyle {\hat {\alpha }}={\underset {\alpha }{\operatorname {arg\,min} }}\,D_{\alpha }}$

with

${\displaystyle D_{\alpha }=\max _{x}|P_{\mathrm {emp} }(x)-P_{\alpha }(x)|}$

where ${\displaystyle P_{\mathrm {emp} }(x)}$ and ${\displaystyle P_{\alpha }(x)}$ denote the cdfs of the data and the power law with exponent ${\displaystyle \alpha }$, respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored.[4]

#### Two-point fitting method

This criterion[clarification needed] can be applied for the estimation of power-law exponent in the case of scale free distributions and provides a more convergent estimate than the maximum likelihood method.[citation needed] It has been applied to study probability distributions of fracture apertures.[citation needed] In some contexts the probability distribution is described, not by the cumulative distribution function, by the cumulative frequency of a property X, defined as the number of elements per meter (or area unit, second etc.) for which X > x applies, where x is a variable real number. As an example,[citation needed] the cumulative distribution of the fracture aperture, X, for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope).

## Validating power laws

Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data.[24] This is important for understanding the mechanism that gives rise to the distribution: superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation.

For example, log-normal distributions are often mistaken for power-law distributions:[53] a data set drawn from a lognormal distribution will be approximately linear for large values (corresponding to the upper tail of the lognormal being close to a power law)[clarification needed], but for small values the lognormal will drop off significantly (bowing down), corresponding to the lower tail of the lognormal being small (there are very few small values, rather than many small values in a power law).[citation needed]

For example, Gibrat's law about proportional growth processes produce distributions that are lognormal, although their log–log plots look linear over a limited range. An explanation of this is that although the logarithm of the lognormal density function is quadratic in log(x), yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law.

In general, many alternative functional forms can appear to follow a power-law form for some extent.[54] Stumpf[55] proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude. Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a power law. As a solution to this problem, Diaz[38] proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails. However, Stumpf[55] claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process.

One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. As such, the validation of power-law claims remains a very active field of research in many areas of modern science.[9]

## References

Notes

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Home | Current | Past volumes | About | Login | Notify | Contact | Search

 Equidistribution, uniform distribution: a probabilist's perspective

 Abstract The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed $$\mathsf{mod}~1$$ sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original $$p$$-multiply equidistributed sequence $$k^p\, t\ {\mathsf{mod}}~1$$, $$k\geq1$$ (where $$p\in\mathbb{N}$$ and $$t\in[0,1]$$), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to $$\infty$$-distributed Koksma's numbers $$t^k\ {\mathsf{mod}}~1$$, $$k\geq1$$ (where $$t\in[1,a]$$ for some $$a>1$$), and an important generalization by Niederreiter and Tichy from 1985 are discussed. The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as certain computer scientists and number theorists. AMS 2000 subject classifications: Primary 60-01, 11-02; secondary 11K45, 65C10, 60F15.Keywords: Equidistribution, completely equidistributed, completely uniformly distributed, ∞-distributed, metric theory, weakly completely uniformly distributed, pseudo-random numbers, Weyl criterion, strong law of large numbers, weakly correlated, dependent random variables.
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Limic, Vlada, Limic, Nedzad, Equidistribution, uniform distribution: a probabilist's perspective, Probability Surveys, 15, (2018), 131-155 (electronic). DOI: 10.1214/17-PS295.

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# Explicitness in numeral system

Prove that for every $a\in\mathbb{N}$ there is one and only one way to express it in the system with base $\mathbb{N}\ni s>1$.

Seems classical, but I don't have any specific argument.

-
The most direct proof sketch, I think, is that there is an algorithm for computing the $k$-th digit of $a$. This statement (as stated) wouldn't even make sense if $s$-adic representation were not unique! To flesh out the sketch, you revisit the argument that the algorithm computes what is stated. –  Hurkyl Aug 16 '12 at 14:22

Suppose that some positive integer $N$ has two representations base $s$. Then we can write $$N=\sum_{k=0}^ma_ks^k=\sum_{k=0}^nb_ks^k\;,\tag{1}$$ where $0\le a_1,\dots,a_m,b_1,\dots,b_n\le s-1$ are integers, and without loss of generality we may assume that $m\le n$.

Suppose first that $m<n$. Then

\begin{align*} \sum_{k=0}^ma_ks^k&\le\sum_{k=0}^m(s-1)s^k\\ &=\sum_{k=0}^ms^{k+1}-\sum_{k=0}^ms^k\\ &=\sum_{k=1}^{m+1}s^k-\sum_{k=0}^ms^k\\ &=s^{m+1}-1\\ &<s^n\\ &\le\sum_{k=0}^nb_ks^k\;, \end{align*}

contradiction the assumption $(1)$. Thus, we assume that $m=n$.

Now make the further assumption that $N$ is the smallest positive integer with two different base $s$ representations. Without loss of generality assume that $a_n\le b_n$, and let $M=N-a_ns^n$. Then

$$M=\sum_{k=0}^{n-1}a_ks^k=\sum_{k=0}^{n-1}b_ks^k+(b_n-a_n)s^n\;.\tag{2}$$

We just proved that this is impossible if $b_n-a_n\ne 0$, so $(2)$ must simplify to

$$M=\sum_{k=0}^{n-1}a_ks^k=\sum_{k=0}^{n-1}b_ks^k\;.$$

But then $M$ is a number with two base $s$ representations that is smaller than $N$, contradicting our choice of $N$. It follows that no such $N$ exists and hence that every positive integer has a unique representation base $s$.

-

We can express a natural number $a$ to any base $s$ by writing $k = \lceil \log_s a \rceil$, the number of digits we'll need, $a_k=\max(\{n\in \mathbb{N}: ns^k\leq a\}),$ and recursively $a_{i}=\max(\{n\in \mathbb{N}: ns^i \leq a-\sum_{j=i+1}^k a_j s^j\})$. It's immediate that each of these maxes exists, since $0s^i\leq a$ no matter what and $a s^i\geq a$, and that they're unique by, say, the well-ordering of $\mathbb{N}$. Then $$a=\sum_{j=0}^k a_j s^k$$

-

We show existence by (strong) induction. Suppose the result is true for all $m\lt n$. Show it is true at $n$.

Let $s^k$ be the largest power of $s$ that is $\le n$. Then $1\le n/s^k \lt s$. So if $d=\lfloor n/s^k\rfloor$ then $s=ds^k +m$ where $0\le m\lt n$ and $m\lt s^k$. If $m=0$ we are finished, else apply induction hypothesis.

For uniqueness, again we can use induction. The lead digit is uniquely determined by the condition in the preceding paragraph.

Another way: Uniqueness is particularly easy to prove by an induction running in the other direction. Take a particular representation, of a possible several. The "units" digit $d_0$ is uniquely determined as the remainder when $n$ is divided by $s$. But then the next digit $d_1$ is the remainder when $(n-d_0)/s$ is divided by $s$. This is determined completely by $n$. But then the next digit is the remainder when $(n-d_0-d_1s)/s^2$ is divided by $s$, again completely determined by $n$. And so on.

Existence can be proved in the same way. Let $d_0$ be the remainder when $n$ is divided by $s$. Then $n=ms+d_0$, and by induction hypothesis $m$ has a representation.

-

This is a variation of "another way" by André Nicolas, proving existence and existence at uniqueness at the same time. The main reason I write this answer is that I feel that any complete proof should explicitly use in the uniqueness part the fact that a leading digit "$0$" is forbidden, and in the existence part that $s>1$. So the statement to prove is:

For $s\in\mathbb N$ with $s>1$ and any $a\in\mathbb N$ there exists a unique sequence $(a_{k-1},\ldots,a_0)$ with $k\in\mathbb N$ and $a_i\in\{0,\ldots,s-1\}$ for $0\leq i<k$, and $a_{k-1}\neq0$ if $k>0$, such that $a=\sum_{0\leq i<k}a_is^i$.

For all $n>0$ one has $sn>n$ (since $s>1$) so for every $m>0$ the quotient $\lfloor\frac ms\rfloor$ of the Eucliedean division of $m$ by $s$ is strictly smaller than $m$; this will allow us to do a strong induction on $a$ in which the induction hypothesis applies to $\lfloor\frac as\rfloor$.

The case $a=0$ needs separate consideration; the empty sequence ($k=0$) is a solution, and it is unique since for any sequence with $k>0$ the requirement $a_{k-1}\neq0$ implies $\sum_{0\leq i<k}a_is^i>0$. Now assume $a>0$ and by strong induction that the unique existence has been proved for $q=\lfloor\frac as\rfloor$. Writing $a=sq+r$ one has $r=a\bmod s$ (the remainder of Euclidean division) and for any solution $a=\sum_{0\leq i<k}a_is^i$, now obviously with $k>0$, one has $a\equiv a_0\pmod s$; therefore $a_0=r$ by the requirement $a_0\in\{0,\ldots,s-1\}$. Given that fact, the condition $a=\sum_{0\leq i<k}a_is^i$ is equivalent to $\sum_{1\leq i<k}a_is^{i-1}=\lfloor\frac as\rfloor=q$, which by the induction hypothesis has a unique solution $(a_{k-1},\ldots,a_1)$. If $k=1$ one has $a_{k-1}=a_0=a\neq0$ by assumption, and if $k>1$ then $a_{k-1}\neq0$ is part of the requirement for a solution for $q$.

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