Wikipedia:Reference desk/Archives/Mathematics/2009 March 10

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 10

Pearson's Square namesake?

Pearson's Square, a method for blending mixtures, is not to be confused with Pearson's Chi Square by Karl Pearson who is not connected to Pearson's Square in any of many searches. So, who is this Pearson Square person?69.72.68.7 (talk) 02:54, 10 March 2009 (UTC)

Dunno, wiki doesn't seem to have an article on it ;-) I have come across this before so I guess it is notable - so why not start an article on it? Dmcq (talk) 23:36, 10 March 2009 (UTC)

Googled a bit, found no original reference. Pearson's Square is used in this 1922 paper, so it must have been well known at that time. Karl was born in 1857 (and 65 in 1922), Egon was born in 1895 (and 27 in 1922). --NorwegianBlue ^talk 20:43, 11 March 2009 (UTC)

Very quick question on constant functions

Hi there - if we have a function $f:\mathbb {R} \to \mathbb {R}$ s.t. $|f(x)-f(y)|\leq |x-y|^{2}\forall x,y\in \mathbb {R}$, I want to show f is constant. I'm tempted to just say $|{\frac {f(x)-f(y)}{x-y}}|\leq |x-y|$ so taking the limit as $x\rightarrow y$ we get f'(x)=0 - however, don't you have to assume differentiability to take this limit? Naturally f(x)=c is differentiable with no problems but it seems somewhat circular to say f(x)=constant because of a property we get from f(x) being differentiable which we know is possible because f is constant...

Does my argument need refining? Where would I start?

Thanks, Otherlobby17 (talk) 08:28, 10 March 2009 (UTC)Otherlobby17

From the inequality, it follows that the limit defining the derivative exists and is 0. Ringspectrum (talk) 08:45, 10 March 2009 (UTC)

Given any sequence x_n converging to y, the sequence of elements of the form Δf (x_n) (where Δf denotes the difference quotient of f) is bounded between two sequences; one sequence being the constant "0" sequence (lower bound) and the other sequence being an arbitrary sequence converging to 0 (upper bound if x_n greater than y for all y (the other case is similarly handled)). By the squeeze theorem, the sequence of difference quotients converges to 0; since the sequence we choose was arbitrary, we have established that the derivative of f at y is 0. --PST 09:17, 10 March 2009 (UTC)

well, in fact the squeeze theorem is also stated for functions, in which case there is no real need of an addendum to Ringspectrum's neat argument ;) --pma (talk) 10:59, 10 March 2009 (UTC)

My argument basically proves the squeeze theorem for functions just in case it was not clear enough for the OP. I like sequences better than functions because they encode enough information when you are living in a topological space(although sequences are actually functions!). --PST 07:46, 11 March 2009 (UTC)

Sequences encode enough information when you're in a topspace? What do you mean by that? Algebraist 20:48, 11 March 2009 (UTC)

He means he lives in a first countable space, I suppose ;) --pma (talk) 08:54, 12 March 2009 (UTC)

Right, but I prefer to allow all topological spaces to be in the zoo rather than just first countable ones. :) --PST 09:23, 12 March 2009 (UTC)

So what did you mean, then? Algebraist 10:21, 12 March 2009 (UTC)

Where did the symbol (J) for irrationals on this page originate from?

Hi,

Question: At the bottom of the following page, the symbol (J) is listed for irrationals with R for Real, and Z for integers. Could someone verify, prove or provide the origination of this information?

URL: http://en.wikipedia.org/wiki/Irrational_number

Thanks, doug

Basic Natural numbers (N) · Integers (Z) · Rational numbers (Q) · Irrational numbers (J) · Real numbers (R) · Imaginary numbers (I)· 12.77.184.121 (talk) 10:20, 10 March 2009 (UTC)

I and J are just local notations; I don't know where their story originated, and where it will end :) --pma (talk) 11:04, 10 March 2009 (UTC)

I've never seen this notation, on Wikipedia or elsewhere. It was added to the template by an anonymous user without explanation in January, and I strongly suspect it is simply bogus, so I'll revert it. — Emil J. 11:11, 10 March 2009 (UTC)

...so now I know where it will end: exactly where I thought ;) --pma (talk) 11:29, 10 March 2009 (UTC) I

I've seen it before, but very rarely. $\mathbb {R} \backslash \mathbb {Q}$ is much more common. --Tango (talk) 11:33, 10 March 2009 (UTC)

Think it might be old-fashioned continental, e.g. (iirc) from Bourbaki General Topology, the section on Baire space (set theory) and Polish spaces.John Z (talk) 19:44, 15 March 2009 (UTC)

User:Anonymous Dissident/Log

I'm working on a new article about differentiation by taking logarithms. It's a bit of spin-off of logarithmic derivative, which mostly concerns the derivatives of logarithms (oddly); this one I've tried concerns the technique of taking the natural log of both sides of some trickier functions in order to simplify them before differentiation.

Because I'm an amateur who merely takes an interest in mathematics, and this is my first article on the subject, I'd not feel right moving it to the mainspace before someone more informed had a look in. It's possible I've made mistakes, in the content and maybe in the calculations. Therefore, I'd appreciate some feedback and some review of how it's looking. At any rate, it's by no means done; there is still referencing to be done, as well as general cleanup. I appreciate the help. Best, —Anonymous Dissident^Talk 15:04, 10 March 2009 (UTC)

Just for the other ref-deskers: the article appears fit for mainspace as far as policy, notability, formatting, sources, wiki links etc. This question really is just about the math content itself, which is first year calculus (taking derivatives quotients, powers, and compositions of elementary functions).

For the OP: If you don't get an answer here, you might try at WT:MATH, the WikiProject Mathematics discussion page. It is probably the right place to ask if you need more assistance. Some of the math is "wrong" or "incomplete", but not horribly so. It needs somebody to check it over and correct a few small things, but overall it looks fine. The references could use some polishing too (author; find non-self published works like a paper/dead-tree calculus textbook). JackSchmidt (talk) 15:24, 10 March 2009 (UTC)

I think the question is whether this violates Wikipedia is not a textbook. I'm not sure this technique is notable, it's just a useful trick for differentiating certain functions. --Tango (talk) 16:27, 10 March 2009 (UTC)

As I said, the referencing is not finished. I have quite a few more reliable sources, and this is quite well documented. It might not be of the same order as the chain or product or quotient rule, but I've definitely seen it used in even textbooks. So, I'm not really sure it's just a trick; that's what I thought at first, but it's wide usage and actual prescription lead me to think it could be classified as a technique. I did try, deliberately, to make sure it was not textbook-y. Perhaps I overdid it on the examples? In that regard, I was just following the suit of articles like chain ruleor product rule. Perhaps I'll tone those down a bit. —Anonymous Dissident^Talk 20:40, 10 March 2009 (UTC)

If you want my personal opinion, I think logarithmic differentiation is noteworthy. I was taught this technique in high-school calculus, so I suspect most high-school graduates, at least in my province (Ontario), know about it. That's notable enough to warrant an article. --Bowlhover (talk) 22:24, 13 March 2009 (UTC)

Note that while Wikipedia is not a textbook, some of Wikipedia's math editors are also working on the Wikibooks b:Calculus project, which is a textbook. Wikibooks is a sister project to Wikipedia, which works on textbooks. The calculus book can always use more contributors. 76.195.10.34 (talk) 20:56, 10 March 2009 (UTC)

curves within curves

What is the volume of a sphere of given radius in hyperbolic 3-space? —Tamfang (talk) 15:47, 10 March 2009 (UTC)

On page 83 of:

• Ratcliffe, John G. (1994), Foundations of hyperbolic manifolds, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94348-0

Exercise 3.4.5 asks to show that the volume of a 3-dimensional hyperbolic ball of hyperbolic radius r is π·( sinh(2r) − 2r ). JackSchmidt (talk) 16:27, 10 March 2009 (UTC)

Thanks. Let me guess: in any dimension, you lop off the beginning of the Taylor series for sinh(a*r) or cosh(a*r) and scale it so that it approaches the Euclidean case for small r? How do you find a? —Tamfang (talk) 21:36, 11 March 2009 (UTC)

That may well work, but something a little more rigorous would be preferred! If that method does work (I haven't tried to prove it), I think you need to choose a so that the coefficient of the nth term (for an n-ball) matches the fraction in the Euclidean case (and then you minus off all the preceding terms), and put a pi out the front. --Tango (talk) 02:29, 12 March 2009 (UTC)

Or some multiple of a power of pi: the question is which factors go inside and which outside. —Tamfang (talk) 04:51, 13 March 2009 (UTC)

Energy loss in Reflection (intensity loss) [formula wanted please]

Move to Science desk. --Tango (talk) 17:02, 10 March 2009 (UTC)

Probability

So, I am trying to figure out the probability that something won't happen. This is in reference to a computer game, World of Warcraft. There is a boss in this game you can kill once a week. This boss has a 17% chance of dropping Item X. This implies an 83% chance to not drop it. So I would assume the chance to not drop 'Item X,' 11 times in a row would be equal to (0.83)^11 = ~0.22 = 22% chance that he won't drop it 11 time straight. My first question is this: is my math correct up to this point?

Second question: It takes 25 people cooperating to kill this boss; Does this affect the drop rate or probability of attainment for an individual? (i.e. 1/25 * 22% to arrive at 0.8% from an individual's perspective.....Does perspective here, either as a group or individual, even play in to the statistics?--Mrdeath5493 (talk) 19:18, 10 March 2009 (UTC)

(0.83)^11 is about 0.13 = 13%; so there is an 87% chance that it drops at least once after 11 runs. The drop rate for an individual depends on how you divide the loot. If everyone has an equal shot, then the probability of one particular person getting it is a little complicated, but is at least 87%/25 which is about 3%. It is a little higher, since it might drop twice or even eleven times. My calculations are that assuming everyone has an equal shot, the odds of an individual getting are:

${\tfrac {1}{25}}\sum _{i=0}^{11}i\cdot {\binom {11}{i}}\cdot 0.17^{i}\cdot 0.83^{11-i}\approx 7.5\%$

This assumes a person still rolls even if they already have it from a previous run. My calculation for if people stop rolling once they get it is a little fuzzier, but only comes out at 7.7%. JackSchmidt (talk) 20:42, 10 March 2009 (UTC)

I noticed that 0.87^11 is about 0.22, so in case the drop rate was 13%, here is the calculation

${\tfrac {1}{25}}\sum _{i=0}^{11}i\cdot {\binom {11}{i}}\cdot 0.13^{i}\cdot 0.87^{11-i}\approx 5.7\%$

and the "stop rolling" version is about 5.9%. JackSchmidt (talk) 20:44, 10 March 2009 (UTC)