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May 13

A slight variation on maps from the disc to the UHP model

Hi all,

I moved this back onto the reference desk because I realise I made a mistake typing up the question, hopefully now it's corrected someone will be able to help! (Sorry if that's bad 'reference desk form', I didn't think anyone would notice any changes that far up the page)

I'm fiddling around with a little bit of mapping in the complex plane, and I'm trying to find an analytic map between G = $\{z\in \mathbb {C} :|z|<\,1,\,|z+i|\,>\,{\sqrt {2}}\}$ and S = $\{x+iy:\,-\pi \,<\,x\,<\,\pi \}$ - I'm used to doing these problems for two circles one inside the other, or two circles meeting at just one point on the boundary - which we can map to infinity quite easily with a mobius map, since boundaries map to boundaries, but in this case I'm having problems seeing where to go because I need to map both $\pm 1$ to infinity (both of these points are on the intersection of the two disc boundaries which comprise G), so I'm no longer looking at mobius maps.

Can anyone suggest what I might try? I'm perfectly comfortable with all the concepts, just haven't ever had to use a map with 2 points at infinity on the boundary of G. I'd really appreciate any help!

Spamalert101 (talk) 12:54, 11 May 2010 (UTC)[reply]

The set G you've described is empty. Did you mean

G=\{z\in \mathbb {C} :|z|<\,1,\,|z+i|\,>\,{\sqrt {2}}\}

? -- Meni Rosenfeld (talk) 19:05, 12 May 2010 (UTC)[reply]

I did indeed, it's now corrected - thanks for spotting that! Does anyone have any suggestions please? Thankyou in advance :) Spamalert101 (talk) 00:50, 15 May 2010 (UTC)[reply]

Is there a name for this special complex function?

Does anybody know whether the inverse function (at z=0) of

f(z):=z(1+z)^α

has a name? (here α∈C).

Thanks, --pm a 08:04, 13 May 2010 (UTC)[reply]

No but substituting x for the αth root of z gives something like the trinomial in http://xxx.lanl.gov/PS_cache/math/pdf/9411/9411224v1.pdf Dmcq (talk) 08:27, 13 May 2010 (UTC)[reply]

Thanks, very interesting link. --pm a 08:55, 13 May 2010 (UTC)[reply]

Pma, I was wondering whether you know about mathoverflow. The reference desk is often an excellent place to receive answers to mathematics questions in general, but for certain sorts of questions, you may be more likely to receive answers at mathoverflow rather than at Wikipedia (for instance, research-level mathematics questions; in fact, the purpose of mathoverflow is to aid mathematicians in this sense). There are definitely many knowledgeable people here at the reference desk, but it seems that most questions asked here are somehow related to homework/learning; mathoverflow is geared toward a different class of mathematics questions (as outlined here). PS T 10:10, 13 May 2010 (UTC)[reply]

Thank you very much PST for this link! Actually my question is related with teaching (the exercise is to write the power series expansion at 0 of that function. In similar cases, I like to call objects with their names, if they have one, so that people can connect what they learn to other facts).--pm a 11:02, 13 May 2010 (UTC)[reply]

The power series expansion of the composition inverse of

f

is

\sum _{n=1}^{\infty }{\frac {1}{n}}{-n\alpha  \choose n-1}z^{n}

, btw. --pm a 20:33, 13 May 2010 (UTC)[reply]

calculation help

Hello, please could you help me understand a statistical comparison. The original article says "A's serum levels were 41% higher than B's". I want to compare B to A instead - would I say B's levels were 29 per cent lower than A's, or is it still 41 per cent? Thank you. —Preceding unsigned comment added by 207.134.250.140 (talk) 13:17, 13 May 2010 (UTC)[reply]

Hello again, I needed this urgently so I asked Dr. Math and they replied. For interest, they agree with 29 per cent and provided a formula: A is 1+x times B, and B is 1/(1+x) times A. The decrease is y = 1-1/(1+x). Signed 207.134.250.140 —Preceding unsigned comment added by 207.134.250.140 (talk) 16:03, 13 May 2010 (UTC)[reply]

Yes, that's correct. Think of A as 141 and B as 100, so B is 41 lower than A. In percentage terms, we express this as a percentage of the starting amount (A in this case), so B is 41/141 times 100% = just over 29% lower than A. Dbfirs 18:26, 13 May 2010 (UTC)[reply]

Thanks for the confirmation! Signed, 207.134.250.140 (talk) —Preceding undated comment added 20:46, 13 May 2010 (UTC).[reply]

The relative increase from A to B (where B>A) is (B−A)/A *(100%), and the (negative) increase from B back to A is (A−B)/B * (100%) = −(B−A)/B * (100%) . They should sum to zero, but do not. So the formula is simply not suitable. A better formula for the relative increase from A to B is log(B/A). The increase from B to A is then log(A/B) = −log(B/A). Regrettably there is no commonly used name for the logarithmic unit corresponding to the percent. So the confusion of the low-quality percentage mathematics continues. Bo Jacoby (talk) 12:38, 14 May 2010 (UTC).[reply]

Yes, I like your "logcent" method. Do you think it will catch on? If we abandon simple percentages, then we would have to stop talking about fractional changes, including "double" and "half". (Double would be a 100 logcent increase [same as percent] and half a 100 logcent decrease if we use log to the base 2). Dbfirs 13:49, 14 May 2010 (UTC)[reply]

Hey, this product usually costs $10, but now it's only $8! That's a discount of... Wait, let me get my calculator.

I agree that a logarithmic scale is useful and elegant, and would like to see it more widely used. I don't agree that it's as easy to use in all cases and should replace the arithmetic scale completely. That would be like saying that the multiplication operation should be abandoned and we will speak only in terms of addition, exponentiation and logarithms.

What's the solution to $\exp(\log a+\log x+\log x)+\exp(\log b+\log x)+c=0\;\!$ ?

-- Meni Rosenfeld (talk) 14:38, 14 May 2010 (UTC)[reply]

Thank you for your comments, and for the fine word 'logcent', but I suggest using natural logaritms. One logcent is then almost the same as one percent, (because e^0.01≈1.01), and 100 logcents is the factor 2.7, (because 1.01¹⁰⁰ ≈ lim_n→∞ (1+1/n)ⁿ = e). Double is 69 logcents, and half is −69 logcents, because log 2 = 0.69. An exchange rate of 91 Japanese yen (JPY, ¥) to the United States dollar (USD, $) means that ¥ = $ − 451 logcent, and $ = ¥ + 451 logcent, because log 91 = 4.51. No Meni, I do not suggest to abandon multiplication, but I do suggest to abandon the use of percentage for relative comparisons and exchange rates. Bo Jacoby (talk) 00:46, 15 May 2010 (UTC).[reply]

Does anyone actually use percentage for exchange rates? I thought everyone just used a multiplication (or division) factor. I don't think we will ever change normal percentages for simple comparisons because many people would struggle with natural logarithms (even more than they struggle with normal percentage increase & decrease). I suppose if everyone was supplied with a scientific calculator with special logcent functions .... Dbfirs 18:32, 15 May 2010 (UTC)[reply]

Quoting the article Price_index#Normalizing_index_numbers: "Price indices generally select a base year and make that index value equal to 100. You then express every other year as a percentage of that base year". (My italics). Still it is a multiplication factor. Percentages for simple comparisons will change when mathematicians criticize the price index industry. The confusing distinction between absolute and relative variation will disappear when we change to a logarithmic index. For example http://www.nasdaq.com/help/helpfaq.stm is quoting both: "May 14, 2010 US Market Closed NASDAQ 2346.85 -47.51 -1.98%". (Note that 100% · (−47.51) / (2346.85+47.51) = -1.98% ). Using a logcent index it should simplify to: May 14, 2010 US Market Closed NASDAQ 315.57 -2.00, (because 100 · log(2346.85/100) = 315.566 and 100 · log (2346.85 / (2346.85 + 47.51)) = -2.0042). Bo Jacoby (talk) 04:16, 16 May 2010 (UTC).[reply]

You may be interested to know that Tim Cole proposed something very similar, if not identical, to what Bo Jacoby has proposed above and Dbfirs has christened 'logcent', but came up with the name 'sympercent': Cole TJ. Sympercents: symmetric percentage differences on the 100 log(e) scale simplify the presentation of log transformed data. Statistics in Medicine 2000; 19(22):3109-3125. doi:10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F --Qwfp (talk) 08:54, 16 May 2010 (UTC)[reply]

Thank you! The unit of natural logarithm is called Np, neper, but the subunit cNp, centi-neper, is not commonly used, if at all. (The word may be confused with centimeter). Astrophysical Quantities (§5 Logarithmic Quantities) introduces the name exp for the unit of the natural logarithm, (and the name dex for the unit of the base 10 logarithm), but the centiexp is not much used, if at all. Then comes Dbfirs' logcent and Cole's sympercent. This matter of naming may not yet be settled. I vote for logcent. The need for a name is obvious to me. The big and small numbers of economy, physics and astronomy are hard to pronounce and grasp. It is nicer to say that one joule is 7640 logcent hertz, than that the Planck constant is 6.626·10⁻³⁴ J s. Bo Jacoby (talk) 15:15, 16 May 2010 (UTC).[reply]

Thanks, I'd never heard of the neper before. Hmm, logcent, sympercent, centineper... how about 'nepercent' ?? --Qwfp (talk) 18:25, 16 May 2010 (UTC)[reply]

The word degree and the symbol ° is sometimes used for units on some scale. What about May 14, 2010 US Market Closed NASDAQ 315.57° -2.00°? Bo Jacoby (talk) 06:57, 20 May 2010 (UTC).[reply]