Wikipedia:Reference desk/Archives/Mathematics/2012 November 16

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November 16

I have question about prices and gains

Hi, well, my questions is this: I sold 75 boxes for a price of $58 a unit and my gain is 15%, what would be my total gain and gain per unit? Thanks — Preceding unsigned comment added by 189.214.112.249 (talk) 00:18, 16 November 2012 (UTC)

What, so you sold $4350? Then ${\frac {\$4350}{1.15}}=\$3782.61$ is how much it cost you. $\$4350-\$3782.61=\$567.39$ is your total gain. ${\frac {\$567.39}{75}}=\$7.57$ is your gain per unit. Of course there are other ways of working this out.--AnalysisAlgebra (talk) 05:18, 16 November 2012 (UTC)

Is this an application of the correspondence theorem?

It might be, but maybe not. Let $\phi :G\rightarrow G'$ be a surjective homomorphism and let $H'\leq G'$ be a subgroup of $G'$. Show that there is a bijective correspondence between $K'\leq G'$ containing $H'$ and $K\leq G$ containing $\phi ^{-1}(H')$--AnalysisAlgebra (talk) 05:32, 16 November 2012 (UTC)

Probability that someone born in 1865 was still alive in 1945

Hello, could someone please show me how to calculate the probability that someone from Austria born in 1865 was still alive in 1945 and in 1950? One may use these pyramids and these data for the calculation. I've tried to do it myself, but have stuck. Is the likelihood of this being true statistically significant per standards used in the population statistics? Thanks a lot. --Eleassar ^{my talk} 07:55, 16 November 2012 (UTC)

I looked at the pyramid. The exact years 1865, 1945 and 1950 are not available. But 1945-1865=80 years and 1950-1865=85 years.

In year 1869 the number of newborns were 114819.

In year 1951 the number of 80 year olds were 17218.

In year 1951 the number of 85 year olds were 6121.

The probability of for a newborn around 1865 to survive until 80 years of age is 17218/114819=15%

The probability of for a newborn around 1865 to survive until 85 years of age is 6121/114819=5%

I don't think that the concept of statistical significance is relevant here. If in year 1945 you did meet an Austrian born in 1865 then you should be able to decide whether he is dead or alive and then the probability makes no sense.

Bo Jacoby (talk) 14:15, 16 November 2012 (UTC).

Thank you very much, particularly for explaining me how this is calculated. --Eleassar ^{my talk} 14:44, 16 November 2012 (UTC)

Does that count migration? – b_jonas 20:48, 17 November 2012 (UTC)

No. Bo Jacoby (talk) 17:20, 19 November 2012 (UTC).

Chaos

If I have an n-dimensional unknown dynamical system ${\dot {\mathbf {x} }}(t)=f(\mathbf {x} (t))$ and I sample an m≤n dimensional linear map $\mathbf {y} (t)=M\mathbf {x} (t)$ at intervals, building up a finite data set {${\mathbf {y} }_{1}\cdots {\mathbf {y} }_{N}$}. Is there anyway of determining whether $\mathbf {x} (t)$ is chaotic or not?

And furthermore, if my sample introduces a stochastic noise term $\mathbf {n}$ so that my sample is taken from the function $\mathbf {y} (t)=M\mathbf {x} (t)+\mathbf {n} (t)$ is there anyway to determine whether $\mathbf {x} (t)$ is chaotic or not?

I thought an estimate of the Hausdorff dimension might be able to uncover this? But I am not sure of my reasoning. — Preceding unsigned comment added by 123.136.64.14 (talk) 08:36, 16 November 2012 (UTC)

See Correlation dimension. The Hausdorff dimension would also work, I presume. Also the BDS test (Brock-Dechert-Scheinkman test) can be used, although we have no article on it.

As I vaguely recall, these tests look for nonlinear determinism (which in a bounded and non-convergent data set would be chaos) against a null of white noise. But in the presence of complicated deterministic systems they may not have very high power to detect it (see note 4 in Correlation dimension). If the data are infected with noise, they'll have even less power. Duoduoduo (talk) 19:57, 16 November 2012 (UTC)

You can measure the Lyapunov exponents of the system. 130.88.99.231 (talk) 16:03, 21 November 2012 (UTC)

Square root of 2

To paraphrase Isaac Asimov, it's better to discover old things that are correct than to discover new things that are incorrect. Thus,I am sure this has been taught before. I am just wondering if it is an ancient approximation or not. ... Just as there are common approximations such as $\pi \approx {\tfrac {22}{7}}$, does anyone ever use $1{\tfrac {5}{12}}\approx {\sqrt {2}}$? I came across it quite accidentally and have never seen it before, yet it seems so simple that it must be fairly common. Is it?    → Michael J Ⓣ Ⓒ Ⓜ 16:33, 16 November 2012 (UTC)

See Square root of 2#Continued fraction representation and Pell numbers.—Emil J. 18:13, 16 November 2012 (UTC)

(ec) The "best" rational approximations of an irrational number come from the partial quotients of its representation as a continued fraction. For sqrt(2) the continued fraction is [1; 2,2,2,2...] (all "quadratic irrationals" have a CF representation that eventually recurs), from which we get the successive approximations 1, 3/2, 7/5, 17/12 (your discovery), 41/29, etc. I don't know whether 17/12 (or any of the others) has historically been used as an approximation. Incidentally, the continued fraction method explains why 355/113 is such a surprisingly good approximation to $\pi$: its CF is [3;7,15,1,292,.. ] (non-recurring). Taking just [3;7] gives the familiar 22/7; [3;7,15] gives the little-used 333/106, and [3;7,15,1] gives 355/113. As the next term is the relatively large 292, going further doesn't make much difference to the approximation. AndrewWTaylor (talk) 18:24, 16 November 2012 (UTC)

This number has certainly played a role, however I can't judge how important it was. There have been tinkerers, not aware of the mathematical background of those numbers, who tried to measure $\pi$ or ${\sqrt {2}}$, and in the latter case, 17/12 was considered to be the correct result. If I remember correctly, this story has been told in Ancient puzzles, but I'm not sure. --KnightMove (talk) 19:23, 16 November 2012 (UTC)

17/12 works very well because 17² = 289 ≈ 288 = 2 × 12². I doubt it has been used musically as an approximation for the ratio of √2 : 1 (giving the equal-tempered tritone) until recently, as 17 is a relatively large prime number in this context. It might have been used as a close mathematical approximation for √2; given that Square root of 2 mentions that ancient Indian mathematicians knew about the approximation 577⁄408 ≈ 1.414215686, one of the approximations from the Pell numbers, they may have known about the smaller 17⁄12 = 1.416 too.

The previous approximation, 7⁄5 = 1.4, has been used in music as the first tritone you get from the harmonic series (the harmonic seventh), although it is quite flat at 582.5 cents and noticeably different from the usually encountered equal-tempered tritone at 600 cents. However, it was probably not used commonly (i.e. as more than just a harmonic or a sound effect) until recently, as apparently music in just intonation didn't use ratios with numbers that had prime factors greater than five until recently. In mathematics, it was probably too obviously inaccurate to be used much. Double sharp (talk) 07:07, 20 November 2012 (UTC)