Wikipedia:Reference desk/Mathematics

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The Wikipedia Reference Desk covering the topic of mathematics.

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.
See also:
Help desk
Village pump
Help manual

February 22[edit]

February 24[edit]

Reference Formula Policy for Exoplanets[edit]

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
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[1] , ...Stefan-Boltzmann constant

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

{ 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. (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. (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[edit]

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. (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. (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\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[edit]

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?[edit]

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; 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
(sorry wasn't logged in). HTH, Robinh (talk) 20:30, 24 February 2015 (UTC)
Thanx! The paper at 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
HTH, Robinh (talk) 21:48, 24 February 2015 (UTC)
Not sure where Wikipedia gives that, but googling 1260 and rubik lead to 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()]
(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[edit]

Help me to understand some mathematical picture?[edit]

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.
PS: The chess thing is just to make easier to me see it and explain the question. (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[edit]

Sample size and confidence levels[edit]

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)
Not necessarily at all. The 95% confidence interval will be much wider with only 50 samples rather than 1000. Dmcq (talk) 16:32, 28 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. (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)
If you want to read articles about the error, read below
Standard_error (talk) 05:22, 2 March 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[edit]

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 (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[edit]

Box Plot[edit]

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.→ (talk) 23:56, 27 February 2015 (UTC)

February 28[edit]


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)

Identify partial differential equation[edit]

These are really two and a half questions.

I stumbled on a partial differential equation:  u=u(x,y), v=v(x,y), u_x v_y - u_y v_x =1.

Question one: How can I google or otherwise search the net for such kind of equations?

Question two: does anyone happen to recognize this as a famous and named equation I can search by name?

Question 2 1/2: u(x,y), v(x,y) rings a (probably wrong) bell, making me think of Cauchy-Riemann equation. I guess this would be a special case of the equation, giving me  u_x^2 + u_y^2 = 1. This looks somehow familiar but I neglected math for far too long to see what that would be. (talk) 19:17, 28 February 2015 (UTC)

I don't know the name of the equation, but in classical mechanics, the left hand side is a Poisson bracket for u and v if x and y were canonical coordinates. --Mark viking (talk) 22:28, 28 February 2015 (UTC)
I have to admit that, on other occasions, I had several utterly unsuccessful tries on getting a grasp on Poisson brackets. I guess there is some tiny ugly heuristic clue that I'm to dump to pick up myself and everyone else is too tired to mention. (talk) 23:28, 28 February 2015 (UTC)
I'd call this the "Jacobian determinant equation", or something like that, and Google finds several useful results for this search term. It is one equation for two unknown functions u,v, so it is underdetermined.
If u+iv additionally solves Cauchy-Riemann, the only solutions will be affine functions u+iv(z)=az+b where a is a complex number of modulus 1.
The equation  u_x^2 + u_y^2 = 1 is called the eikonal equation. —Kusma (t·c) 12:19, 2 March 2015 (UTC)

Regression with a graph (machine learning)[edit]

Suppose I have a graph of nodes and connections between nodes. The independent variables are this graph and other numerical values assigned to each node. The dependent variable is a point value for the node. How could I use regression to predict the point value of new nodes, based on their connections in the network? Is regression even the right tool to use here? (talk) 20:55, 28 February 2015 (UTC)

Just trying to understand the problem here. So the nodes on the graph each have a value, and that value is in some way based on the other numerical values assigned to that node and nearby nodes, right ? If the relevant "graph distance" is simply the smallest number of nodes to get to each node, then you might use that in a second regression analysis, after first doing a regression analysis without considering nearby nodes.
For example, let's say each node's dependent value is 90% based on the (single, in this example) independent value of that node, and 10% based on the node(s) one step away. I think it might be better to look at as few variables at a time as possible. I think you would be more likely to find convergence that way.StuRat (talk) 21:19, 28 February 2015 (UTC)
Yes, that's a good statement of the problem. (talk) 21:28, 28 February 2015 (UTC)
This sounds indeed like a regression problem. However "regression" is a name for a type of problems, not a specific tool, so you can't "use regression", you need to choose which regression to use.
Usually, either the problem is simple enough that a standard cookie-cutter technique can handle it, or you have a huge search space and you need human intelligence and domain knowledge to constrain it. Your question seems to belong to the second category - so without a description that is less abstract, I don't think we can really help with a solution that will actually give good results for the problem you have at hand. -- Meni Rosenfeld (talk) 10:47, 1 March 2015 (UTC)

Statistical tests for normality[edit]

If the Kolmogorov-Smirnov and Shapiro-Wilk give significantly different results, how does one resolve this ambiguity? All the best: Rich Farmbrough21:50, 28 February 2015 (UTC).

It is not really an ambiguity. As the articles you pointed to state, these are different tests, with Shapiro-Wilk giving greater power for a test of normality and Kolmogorov-Smirnov being a nonparametric test that could be applicable to many different kinds of distributions. So it is entirely possible that the tests give different results. Without more details, it is hard to say anything more specific. --Mark viking (talk) 22:14, 28 February 2015 (UTC)
My "go to" first test is Jarque-Bera, because it's so intuitive - read the instructions carefully, as you need to be aware of the sample size - but it's certainly not the best. It's usually a good place to start, as you just drop the data into Excel and look at the skewness and kurtosis immediately, and those values on their own are a useful place to start thinking about the data. If you're using K/S you may have accidentally re-invented the Lilliefors Test. My personal preference would be to prefer S/W to K/S, but see our articles on the weaknesses in both tests. Where I have found K/S very useful is in fitting general stable distributions: though it takes some care and attention you might use this approach to check the stability parameter. RomanSpa (talk) 23:40, 28 February 2015 (UTC)
Thanks, both, for the replies. I certainly draw some warm fuzzies from the replies, and will look into these tests in a little more depth. All the best: Rich Farmbrough23:45, 28 February 2015 (UTC).
(My last edit got lost in an edit conflict, so I'll try to repeat it...) Re: fitting stable distributions: it's not a graceful process! Re: other tests: I had a colleague who swore by the Epps-Pulley test, which we don't have an article on; the raw reference is "Epps, T. W., and Pulley, L. B. (1983). A test for normality based on the empirical characteristic function. Biometrika 70, 723–726". I haven't used E/P myself for a long time, but I do remember it was a pain in the ass to code up! E/P leads naturally to the BHEP test, which I don't really know, but have heard mildly positive comment on. I suspect it's also a nuisance to code up, alas! RomanSpa (talk) 23:56, 28 February 2015 (UTC)
There's obviously one other point, which we should really have put first: think about the data in practical terms, and ask yourself whether the underlying experiment is one that's likely to produce a normal distribution. Remember that what you're doing is modelling reality. There is always a model, so think about what the reality is that you're modelling: is a normal distribution a plausible outcome from the imagined/theoretical mechanism of the experiment. RomanSpa (talk) 00:04, 1 March 2015 (UTC)

March 1[edit]

March 2[edit]

Find the missing perspective[edit]

This should be a fairly straightforward problem in 3D representation of 4D objects, but I'm having no luck in wrapping my head around this. At first I thought I could treat each as a cube within a 4-cube (since there should be eight of those), with each cell being adjacent to a corresponding cell of the cube, but I can't get the seven below to line up right, so I now believe that I'm thinking about the problem wrong (i.e., I shouldn't think of it as an unfolded tesseract but as a 2x2x2x2 something being projected in some other way). Any thoughts or advice would be greatly appreciated. I've exhausted all that I can really think of to solve this sort of problem, since it doesn't seem to fold the right way - assuming that's the right way to think about this.

Given the seven 3D projections of a hyper-object, provide the eighth:

(Each █ is a cell, with the left side being one layer and the right being the other layer of the 2x2x2 slice.)

_ _
_ _ _ _
_ _ _
_ _
_ _
_ _ _
_ _
_ _
_ _ _
_ _ _
_ _ _
_ _
_ _
_ _ _

Spability951 (talk) 02:39, 2 March 2015 (UTC)

Since there are only 4 dimensions to squish, if they're listing 8 different projections, they must be including projecting onto opposite faces as different projections. The projections onto opposite faces would be reflections of one another, and indeed 1 is a reflection of 5, 2 is a reflection of 7, and 3 is a reflection of 6. So that leaves 4. I don't see a pattern to how they're doing the reflections, so maybe any reflection of 4 will do? (talk) 02:13, 3 March 2015 (UTC)

Finding a proper primitive polynomial[edit]

I'm interested in the Massey-Omura cryptosystem at the moment and I need a primitive polynomial for GF(2^{256}) because I want to use a 256-bit long key. Is there a list I can look at or a webpage with an appropriate calculator our there for me to obtain such a polynomial? — Melab±1 06:54, 2 March 2015 (UTC)

(As I mentioned last time, you could use nimber arithmetic; then you don't need a primitive polynomial.) -- BenRG (talk) 07:44, 2 March 2015 (UTC)
It's unclear to me, though, if the polynomial mask is applied after every multiplication/squaring or not. Also, the concept of nimbers isn't made clear enough in the article for me. — Melab±1 20:04, 2 March 2015 (UTC)
This page has an example primitive polynomial for \mathrm{GF}(2^{256}). -- (talk) 02:35, 3 March 2015 (UTC)


About this integral \int_a^b \! f(x)g(x)\,dx/\int_a^b \! g(x)\,dx

Can be equal

\int_a^b \! f(x)\,dx

Of course the two expressions can be equal, but they usually aren't. -- Meni Rosenfeld (talk) 17:24, 2 March 2015 (UTC)
Consider an extremely simple example:
 f(x) = 1, g(x) = x
\int_a^b \! f(x)\,dx = b-a
\int_a^b \! g(x)\,dx = \int_a^b \! f(x)g(x)\,dx = (b^2-a^2)/2
\int_a^b \! f(x)g(x)\,dx/\int_a^b \! g(x)\,dx = 1
Tamfang (talk) 03:18, 3 March 2015 (UTC)