Wikipedia:Reference desk/Mathematics

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Welcome to the mathematics reference desk. Want a faster answer? Main page: Help searching Wikipedia How can I get my question answered? Provide a short header that gives the general topic of the question. Type ~~~~ (four tildes) at the end – this signs and dates your contribution so we know who wrote what and when. Post your question to only one desk. Don't post personal contact information – it will be removed. We'll answer here within a few days. Note: We don't answer (and may remove) questions that require medical diagnosis or legal advice. We don't answer requests for opinions, predictions or debate. We don't do your homework for you, though we’ll help you past the stuck point. How do I answer a question? Main page: Wikipedia:Reference desk/Guidelines The best answers address the question directly, and back up facts with wikilinks and links to sources. Do not edit others' comments and do not give any medical or legal advice.

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February 23

Liquid flow

Consider liquid emanating from a corner of a flat square tray with a containing wall at the edges. If the outflow and depth are constant the liquid will spread on a quarter-circular front with the radius increasing at a rate proportional to the square root of the elapsed time, until it has reached the two adjacent corners. What will happen then, I.e. What shape will the front take and how will it move with time until the whole surface is covered?→31.54.246.96 (talk) 17:49, 23 February 2015 (UTC)

Have a look at wave propagation, maybe check out this video [1]. Also consider doing some empirical tests. That is probably more fun than setting up a thin-film navier stokes equation and trying to solve it ;) SemanticMantis (talk) 18:04, 23 February 2015 (UTC)

The viscosity of the liquid (usually temperature dependent) will matter, as will it's surface tension and other properties. Other factors are the rate the liquid is added and the tray material and shape (whether there are rounded corners and edges). A thicker fluid poured slowly will tend to spread as you described, while a thin liquid poured quickly will splash around, exhibiting turbulent flow instead of the laminar flow you described. In any case, there will be waves generated off collisions with the walls and corners, although those waves may be quite small for certain conditions. Note that laminar flow is quite predictable, while the actual path taken by the liquid under turbulent flow each time is chaotic, and thus unpredictable, other than as probabilities.

Also, under the laminar flow you described, you may find that the liquid expands more slowly adjacent to the walls, so you don't get a perfect quarter circle. StuRat (talk) 20:30, 23 February 2015 (UTC)

I think the previous respondents have over-complicated the issue. This looks like a homework question to me. Observe that the questioner explicitly notes that the depth of the liquid is constant. This simplifies the problem to simply requiring that the increase in area covered is a constant. So from the point at which the two corners are reached the liquid will simply continue to expand along a front that is a steadily narrowing arc of an increasing circle. If the square is of unit side, the circle will reach maximum radius of sqrt(2), with the front moving faster the closer it gets to the final corner of the square. RomanSpa (talk) 21:04, 23 February 2015 (UTC)

It's not homework, just something that I was trying to get an answer to out of interest. I'm long past the age of taking tests and exams, nor have I ever worked with the complications referred to in the first two responses. Taking the much simpler case referred to the third response, is it possible to get an explicit relation between radius of front and time after reaching the corners adjacent to the source? My analysis led to t being the integral of r(π/4-arccos(1/r)), ignoring a scaling factor, which I couldn't evaluate but which seemed unlikely to lead to r as a function of t.→31.54.246.96 (talk) 23:12, 23 February 2015 (UTC)

Another way of describing it would be viscous fluid being injected between two sheets of glass, with the air freely escaping somehow. The density, surface tension and dynamic effects other than viscosity are all to be neglected. The only facts needed are the geometry and the fact that the viscosity is uniform and dominates such that all other effects (inertia etc.) can be neglected. You would have a scalar field describing the pressure at every point, the flow vector field, and a function giving the boundary's shape. This will produce something along the lines of a partial differential equation, with a nonlinearity at the boundary. Intuitively, the boundary touching the square will grow faster than the central part once the second pair of edges is reached. I expect that numerical modelling would deal with this most easily. —Quondum 01:18, 24 February 2015 (UTC)

Random walk

I have a random walk variable, call it x, each step moving up by 1 with a probability of p and -1 with probability of 1-p. I also have another variable y that is defined as y_k=max{x_1,x_2...x_k}-x_k. How do I get the expected value of y and does it depend on k? Thx 93.136.7.126 (talk) 19:08, 23 February 2015 (UTC)

Well it definitely depends on k. Consider k=1, then y_k=0. But in general y_k need not equal zero. SemanticMantis (talk) 20:08, 23 February 2015 (UTC)

That's true, how stupid that I didn't think of that :) I've ran some Monte Carlo sims since; y_k should obviously diverge to infinity if p<0.5, as random walk then tends to negative infinity, and probably also for p=0.5. It appears to converge quickly (no apparent tendency at k=10^3 ~ 10^5) to n/(p-0.5), with n≈0.21. That's a good enough result for me, but I'd still be interested in the math to calculate it exactly, if anyone knows how. 78.0.233.239 (talk) 02:42, 24 February 2015 (UTC)

Yes I think your reasoning is correct for p not equal to 0.5. As for the analytic calculation, I'm not optimistic that there is an easy way to get this. It can be tricky to get distributions for maxima of fixed lists, and growing lists will be much harder. Anyway, here's some decent refs on simple random walks [2] [3] [4] [5], they may have tidbits that help you work it out. SemanticMantis (talk) 15:07, 24 February 2015 (UTC)

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 = 5.670373(21)×10⁻⁸ W m⁻² K⁻⁴ , ...Stefan-Boltzmann constant

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

${ Solar Constant } = \left( { R }_{ \bigodot }^{ 2 } \sigma { { T }_{ \bigodot } }^{ 4 } \right) / D^{ 2 }$
${ Solar Constant } = \left( \left( { 0.0850 }^{ 2 }\right) \left({5.670373 } E^{ -8 }\right) \left({4715}^{4 } \right) \right) / 0.26^{ 2 } = 1683.678$ $W/m^{ 2 }$
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)

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.[6] 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 [7] 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 $A\le2.2195\ldots,$ then there is a sofa of area A that can pass.
• If $2.2195\ldots < A \le 2.8284\ldots,$ 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)

February 25

Help me to understand some mathematical picture?

Help me to understand something.

Wikipedia has this picture.

toroidal picture

If we imagine a toroidal chess board, A1 would be conected to A8 and A8 conected to A1, B1 would be conected to b8, A1 would be conected to H1,........

Now, how this picture would works (the sperical one)?
I am not fully getting the idea.
http://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/SphereAsSquare.svg/240px-SphereAsSquare.svg.png
PS: The chess thing is just to make easier to me see it and explain the question.201.78.165.137 (talk) 11:24, 25 February 2015 (UTC)

Along the red lines, a7 is connected to b8, a6 to c8, a5 to d8, all the way to a1 connected to h8. Along the blue lines, a1 is connected to h8, b1 to h7, c1 to h6, etc.

Looked at another way, if you take the left-top half of the depicted square, you can fold it to create a cone, with a seam along the red A line. Likewise you can fold the bootom-right half into a cone. Putting the two cones together gives you a surface homeomorphic to a sphere. -- Meni Rosenfeld (talk) 18:51, 25 February 2015 (UTC)

Or, more graphically, take a square sheet of thin, stretchy rubber, fold over diagonally (fold runs from top left to bottom right in the diagram) to make a triangle, seal the open edges – then simply inflate like a balloon into a spherical shape. —Quondum 19:36, 25 February 2015 (UTC)

Right, and for the first picture of a torus, that's like taking your sheet of rubber, rolling it into a tube, then gluing the circular edges together. This leaves us with the (shell of) a donut, the typical picture of a torus. Also it's worth pointing out that the sphere-like thing you construct out of rubber is topologically a sphere, but of course not geometrically a sphere, because your balloon will have pointy bits, and a geometric sphere does not. SemanticMantis (talk) 22:26, 25 February 2015 (UTC)

One can get rid of the pointy bits through a suitable continuous deformation during inflation. One can say that it is homeomorphically (as noted by Meni, and equivalent to topologically) but not diffeomorphically equivalent to a sphere. (But then, if one chose, one could pre-shape the sheet so that the sealed shape has no pointy bits and is diffeomorphically equivalent; it just wouldn't start as a polygon.) —Quondum 22:50, 27 February 2015 (UTC)

February 26

Sample size and confidence levels

I know that logically I should have the same confidence in a statistical test that uses n=50 and gives a 95% confidence level as one that has n=1000, also with a 95% confidence level. But my gut feeling is to trust the one with the bigger sample size more. Is there any basis for this feeling? Bubba73 ^{You talkin' to me?} 09:31, 26 February 2015 (UTC)

No there's no good reason for that though someone might spot a problem with the prior hypothesis with the larger sample. For the test with n=50 the and 95% confidence if you looked at the figures you'd probably think the difference should be blindingly obvious whereas for n=1000 your intuition would say it was still iffy. So you'd probably have the exact opposite gut feeling if you actually looked at the raw data. Dmcq (talk) 11:33, 26 February 2015 (UTC)

One little thing might matter - an error in one data point out of 50 is more likely to change the conclusion than the error in one data point out of 1000. Bubba73 ^{You talkin' to me?} 23:03, 26 February 2015 (UTC)

You do know that statistics from a sample size of n=50 is different from statistics from a sample size of n=1000 EVEN IF THE CONFIDENCE LEVEL IS EXACTLY THE SAME!!! The statistics from n=1000 has a smaller uncertainty or error interval than the one from n=50.

Just because two statistics have the same confidence level DOES NOT MEAN they have the same error interval. Naturally you want the result from the statistics with the smallest error interval. You would be a fool to choose n=50 over n=1000 unless you do not care about the error interval or if the cost of gathering a sampling point is very very expensive. 175.45.116.60 (talk) 00:55, 27 February 2015 (UTC)

I guess that is what I was getting at - the error interval. Is there an article that talks about the error interval? Bubba73 ^{You talkin' to me?} 00:14, 28 February 2015 (UTC)

I would say that one of the underlying issues is the following. Whenever you are applying a statistical test, you are generally going to be making some assumption about the underlying distribution of the data. For example, you might assume the expected values should reflect a constant plus random noise drawn from a normal distribution. When you calculate a 95% threshold you are essentially saying, given the model I expect, how much confidence do I have that my observations conform to that model. However, in the real world, statistical models often prove to be inexact. You might assume random variations that follow a normal distribution, but the truth is a Laplace distribution or something else. If statistics shows that your data doesn't fit the model, is that because you have discovered a physically important signal, or because your understanding of the background noise wasn't very good? With small numbers of data points, one often has to implicitly assume that the underlying model is reasonable (e.g. normally distributed errors), but when you have lots of data you can often test those assumptions and justify more rigorous conclusions. Dragons flight (talk) 00:42, 28 February 2015 (UTC)

Calculation method help

Need help with correct method for calculating this:

I have membership id (which might have multiple members in it) and member id which represents an individual member of an account. I am trying to calculate average deposit / deposit date for memberships as well as for individual members.

example table below:

Membership ID	Member ID	Deposit Date	Deposit Amount
121	1	23-04-2013	500
121	2	07-04-2013	500
131	46	23-04-2013	100
121	1	01-06-2013	900
131	46	01-06-2013	340
541	91	23-04-2013	500
679	51	23-04-2013	500
679	1	23-04-2013	500

— Preceding unsigned comment added by 203.63.22.226 (talk) 14:11, 26 February 2015

I've answered at the same question on the miscellaneous desk. Please don't post the same question on more than one desk. Dbfirs 23:23, 26 February 2015 (UTC)

February 27

Box Plot

How does one draw a box and whisker plot based on a set of numbers ordered from least to greatest and divided into four quartiles?Ohyeahstormtroopers6 (talk) 22:10, 27 February 2015 (UTC)

Our article Box plot seems clear enough.→31.54.246.96 (talk) 23:56, 27 February 2015 (UTC)

February 28

Median

When, in a data set, there are multiple numbers which together make up the median(i.e. 3,6,9,10,12,11), what do you do with those two numbers to determine the median?Ohyeahstormtroopers6 (talk) 01:03, 28 February 2015 (UTC)

As illustrated in the lead of the article Median, for an even number of data points in the data set, the median is the mean of the centre-most pair of data points. —Quondum 03:03, 28 February 2015 (UTC)

Thank You. 2602:306:C541:CC60:6866:CFB1:2D1B:5526 (talk) 05:38, 28 February 2015 (UTC)