Wikipedia:Reference desk/Archives/Mathematics/2015 February 24

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February 24

Reference Formula Policy for Exoplanets

Proxima Centauri#References

This reference section has a formula in it, which in most cases would be considered Original Research and purged from the article ASAP. But apparently there are exceptions to the rule/policy. What I don't understand is why Wikipedians continue to allow Wikipedia to look foolish with articles that claim many newly discovered planets are a "twin of earth" when there are simple formulas, just like the one used in the Proxima article, that show the solar constant or Flux (Irradiance/Isolation) of the planet. Article after article with new planets in the "habitable zone" that are actually receiving much more heat than Venus or much less heat than Mars, but since they are very technically in the "habitiable zone," editors over look that and reference article that call it a twin of Earth.

Planetary equilibrium temperature#Calculation for extrasolar planets

Another well established formula
$T={\sqrt[{4}]{\frac {(1-a)S}{4\epsilon \sigma }}}$
where Wikipedians can just plug in the numbers is at the article subsection Zero Dimensional Climate Models

I know some articles are showing the flux received in the planet's stats box but it should be a policy.
It should be standard if the Semi-major axis and the Luminosity (or Radius & Temperature) of the star are known.

f = L/d²
...or...
f = [(R²)(sbc)(T⁴)]/d²

${L}_{\bigodot }=\left(4\pi {{R}_{\bigodot }}^{2}\sigma {{T}_{\bigodot }}^{4}\right)$
σ = 5.670374419...×10⁻⁸ W⋅m⁻²⋅K⁻⁴ , ...Stefan-Boltzmann constant

because ${L}_{\bigodot }=\left(4\pi {f}{d}^{2}\right)$
then...

${SolarConstant}=\left({R}_{\bigodot }^{2}\sigma {{T}_{\bigodot }}^{4}\right)/D^{2}$
${SolarConstant}=\left(\left({0.0850}^{2}\right)\left({5.670373}E^{-8}\right)\left({4715}^{4}\right)\right)/0.26^{2}=1683.678$ $W/m^{2}$
An example in HD 85512 b which shows it receives more heat than Earth, 1366.078/1683.678 = 123%.

Can someone explain how I can start a committee or a policy review or whatever it takes to solve this problem?
So that we don't continue to see these articles that get away with false claims of discovered Earth twins.

24.79.36.94 (talk) 02:32, 24 February 2015 (UTC)

It's not clear what your question is. The section you refer to contains a formula stating that the density is mass divided by volume. That does not seem to be an especially problematic calculation (calculating the density of Proxima Centauri does not seem very controversial). I do not see the "Article after article with new planets in the 'habitable zone'" that you refer to, and would need to see clearer examples of the questionable content. To be sure, we should not be injecting our own opinions on the "habitable zone" exclusively using formulas from other articles, but rather should only do so if other sources concur. In that case, it may or may not be appropriate to include calculations: Wikipedia should only summarize what reliable sources have to say on the matter. Unfortunately, part of the WP:NOR policy means that we cannot usually undermine the results of published sources with our own calculations, even if those sources turn out to be wrong. Sławomir Biały (talk) 15:14, 24 February 2015 (UTC)

Calculating the Flux received by a planet is not problematic either. As I've illustrated it can be done with two or three known values.
You want an example of planets that aren't what the are supposed to be, Gliese 581 c used to have references saying it is a Earth-like,
where as now it's more truthfully saying it's more likely a Super-Venus. An extra solar planet article should certainly never start like this
There is no reason why the "Planetbox" shouldn't contain the Flux by now,
as other stats boxes (eg. Kepler-186f) are starting to include them for other exoplanets.
This list of "Confirmed small exoplanets in habitable zones" is one that can be checked for misleading suggestions.
Kepler-186 f could receive as little as 10% of the heat the Earth receives and the Planet Characteristics portion of the stat box don't add up.
It says 41% while the Equilibrium Temperature is -30°C. Kepler-438b is another one with contradictory stats,
"announced as being located within the habitable zone of Kepler-438." Where as at every point in its orbit it receives much less heat than Mars.
To me it simply a matter of stating the mathematical facts, rather than only speculations of Astronomers.
The question is in the title.
24.79.36.94 (talk) 23:21, 24 February 2015 (UTC)

We can only report what references say. We cannot right the wrongs of other scientists here. If you want to do that, you have to publish these calculations elsewhere first, and explain why they contradict the "speculations" of published experts. Sławomir Biały (talk) 16:04, 28 February 2015 (UTC)

Moving sofa problem

I don't really understand the upper bound number of this. upper bound shape supposes to look like this. So the highest lower bound we know right now is 2.219, which means anything bigger than that won't go through the hallway. Then how can the upper bound be 2.82? According to the definition in the paper, the lower bound is the largest area that can go through the hall, and the upper bound is the lowest area that cannot go through the hallway. If the upper bound is 2.82 then an area smaller than that ought to be able to move through the hallway then how can the lower bound is 2.219? There is quite a contradiction here, or I'm missing something. The upper or lower bound should be revolving around a number with lower bound is < that number and upper bound is > that number.146.151.84.226 (talk) 04:18, 24 February 2015 (UTC)

Your external link doesn't work for me but I suspect you misunderstood it or it was talking about something else. I have reverted your edits to the article.[1] If x is an unknown number such that it is known that a ≤ x ≤ b, then a is called a lower bound for x, and b is called an upper bound. It's possible for a and b to be far from the actual value of x. PrimeHunter (talk) 04:38, 24 February 2015 (UTC)

That did not answer my question at all. I did not misunderstand it. I'm missing something that I can't see yet, and I need someone who is a math expert to help explain. Your inequality a ≤ x ≤ b is pretty much like what I defined by words above. a is the largest value close to x that we know where as b is the smallest value close to x. I'm a math major here. I do know what I'm talking about. I just don't understand how the lower bound and upper bound have the values they are in the article right now.146.151.84.226 (talk) 04:51, 24 February 2015 (UTC)

You write: "So the highest lower bound we know right now is 2.219, which means anything bigger than that won't go through the hallway." No, a larger couch may pass. Its been shown that the largest sofa possible that can pass (which is unknown) is larger than or equal to the lower bound 2.219 (and smaller than or equal to the upper bound). -Modocc (talk) 05:35, 24 February 2015 (UTC)

Here is a free version of your source. It says "The lower-bound sofa is that sofa which can be moved through the hallway with continuous transformations, while the upper bound sofa cannot be moved through the hallway." It was speaking about two specific sofas when it said "the". Many sofas may at different times or contexts be called lower bounds and upper bounds. "smallest" and "largest" in [2] is your own wrong invention and I have reverted it. Lower bound and upper bound are common math terms and not something special for the moving sofa problem. It would be odd to add definitions of such common terms there. Any size which is known to be possible can be called a lower bound, but if we say "the lower bound" without further context then it will usually be implied to be the best known lower bound, i.e. the largest number which has currently been proven to be possible. PrimeHunter (talk) 06:10, 24 February 2015 (UTC)

The moving sofa problem is an open problem. We don't know the complete solution. Here is what we do know:

• If $A>2.8284\ldots ,$ then there is no sofa of area A that can pass.
• If $A\leq 2.2195\ldots ,$ then there is a sofa of area A that can pass.
• If $2.2195\ldots <A\leq 2.8284\ldots ,$ then we don't know. Maybe there exists a sofa of area A that can pass, maybe not.

The bounds mentioned are not bounds on the set of areas for which a sofa exists. We don't know what this set is, because it's an open problem, so we don't know what are the bounds on this set (well, the lower bound is known to be 0). The bounds are on the set of numbers which, for all we know, could be the sofa constant (defined in the article as the area of the largest sofa that can pass).

"For all we know" is not really a mathematical object, so these should be understood to refer colloquially to our knowledge on the problem (as outlined above). If someone finds a larger sofa that passes, the lower bound on our knowledge will increase. If someone proves that a whole new bunch of areas are impossible, the upper bound will decrease. If someone finds a sofa that passes, and proves that no larger sofa can pass, then the upper and lower bounds will be the same, meaning we know the sofa constant exactly, and the problem will have been solved completely.

Note also that we're always talking about "There is a sofa of area A" and not "all sofas of area A". Even with a very small area, a sofa which is very long and thin will not pass. Also, if there is a sofa of area A, then for every smaller area there is also a sofa, since we can shrink the original. This is why "the set of possible areas" is an interval starting at 0. The upper bound of this interval is the sofa constant, which is unknown.

On a more general note, I don't see why you would think the numbers for these bounds don't make sense. A lower bound is always no greater than an upper bound. So what's wrong with the lower bound being 2.2195 and the upper bound being 2.8284? -- Meni Rosenfeld (talk) 11:34, 24 February 2015 (UTC)

Statistics percentile question

My daughter is taking statistics in college. On a test she took, the question is "The test scores of 32 students are listed below. Find the 46th percentile." 32 * 0.46 = 14.72. The 14th score is 66 and the 15th score is 68. The answer choices were: 68, 14.72, 15, and 67. She answered "67" but the correct answer was 68. Isn't 67 as good of an answer as 68? Bubba73 ^{You talkin' to me?} 04:46, 24 February 2015 (UTC)

No, her text book has presumably defined Percentile#Definition of the Nearest Rank method. It must be a number in the list. The answer choices are carefully chosen to see whether she can correctly apply the definition. At lower education levels you may be able to make guess because only one value is plausible but at college that usually doesn't fly. PrimeHunter (talk) 14:30, 24 February 2015 (UTC)

Her book does discuss that method (and others, I think). But it defines percentile: "A number that divides ordered data into hundredths; percentiles may or may not be part of the data. ...". and the test just says "find the indicated measure". Bubba73 ^{You talkin' to me?} 17:23, 24 February 2015 (UTC)

If this was on a test with 68 as the correct answer then I suspect she has at some time been taught Percentile#Definition of the Nearest Rank method, or something equivalent. But it's possible the test assumes a definition the students have not been taught. PrimeHunter (talk) 17:40, 24 February 2015 (UTC)

It seems to me that your daughter was wrong. I can never remember (and sometimes not even understand) the formal definitions you get in textbooks. But "46th percentile" means a score which 46% of the class scored better than and 54% scored worse than. This fell inside one student, specifically the 15th student, who scored 68.

Maybe my argument will be made clearer by a simple example. Suppose there were just three students, who scored 40, 70, and 80. What is the 50th percentile? What is the 46th percentile? Maproom (talk) 23:41, 25 February 2015 (UTC)

Rubik's Cube Cycle for FRT?

How long is the cycle for turning the Front Clockwise, then the Right Clockwise and then the Top Clockwise? (i.e. how many set of FRT to get back to the solved cube.) I've also asked on Talk:Rubik's Cube because there are some other somewhat similar questions there that have been answered.Naraht (talk) 17:43, 24 February 2015 (UTC)

It might be worth searching or asking on math.stackexchange.com; for example this and this are questions about the order of elements that consist of two basic moves, rather than the three in your question. AndrewWTaylor (talk) 18:16, 24 February 2015 (UTC)

Sage gives 80

 a=CubeGroup()
 (a.F()*a.R()*a.U()).order()
 80

(sorry wasn't logged in). HTH, Robinh (talk) 20:30, 24 February 2015 (UTC)

Thanx! The paper at https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf says the greatest Order of any sequence is 1260, but doesn't indicate what that is.Naraht (talk) 21:34, 24 February 2015 (UTC)

That is a nice resource which I haven't seen before. Thanks! There are many elements of order 1260. Wikipedia gives

(a.R()*a.U()^2*a.D()^(-1)*a.B()*a.D()^(-1)).order()
1260

HTH, Robinh (talk) 21:48, 24 February 2015 (UTC)

Not sure where Wikipedia gives that, but googling 1260 and rubik lead to https://www.speedsolving.com/forum/showthread.php?23185-Possible-orders-of-Rubik-s-Cube-positions which said that R F2 B' U B' is one of the minimal ones which is the same "length" as yours by the standard metrics.Naraht (talk) 22:01, 24 February 2015 (UTC)

It's at Rubik's cube group#Group structure. Actually I'm surprised that neither element of the two-element generating set given by gens_small() has this maximal order:

i.order() for i in a.gens_small()]
[24,60].

(and come to think of it, how come neither of these has a factor of 7, when the order of the whole group has a factor of 7?) Best wishes, Robinh (talk) 23:06, 24 February 2015 (UTC)