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November 10

Conway puzzle

In conway puzzle, the diagram is wrong. This was pointed out 2 years ago, but nothing has happened. Does anyone here have the ability (and inclination) to redraw it? -- SGBailey (talk) 07:36, 10 November 2012 (UTC)

I think the diagram is actually correct, and that it is being misperceived. Looie496 (talk) 15:50, 10 November 2012 (UTC)

A slightly different angle would make it more obvious. the middle block touches the other two blocks at the corners. If you can describe it better then try editing the article to give a short description under the diagram. Dmcq (talk) 15:59, 10 November 2012 (UTC)

I suggest adding a normal projection onto the two vertical planes, to make it's location obvious. StuRat (talk) 20:52, 12 November 2012 (UTC)

The nearest block occupies positions (1,1,1), (2,1,1) and (3,1,1). The vertical block occupies (4,2,2), (4,2,3) and (4,2,4). The farthest block occupies (5,3,5), (5,4,5) and (5,5,5). The vertical block looks like it starts at (5,3,1) but it's an illusion - it starts at (4,2,2), because it is higher it looks from this perspective as if it's farther. -- Meni Rosenfeld (talk) 19:14, 13 November 2012 (UTC)

Euler axis/angle to Tait–Bryan angles (yaw, pitch, roll) conversion

I have a rotation vector for a camera where the length represents the angle and the direction the axis about which I rotate. While this is an elegant way to store rotation it is not very user friendly. One typically wants to know the yaw (pan), pitch (tilt) and roll of the camera. The article on rotation formalisms in three dimensions gives a bunch of conversion formulae for some cases, but not this one. I preferably do not want to go via expressing the rotation as a matrix (combining the first two conversions given), that seems cumbersome if both representations use just 3 values. 41.164.7.242 (talk) 13:20, 10 November 2012 (UTC) Eon

Axis-angle representation and Rodrigues' rotation formula may well help you here as they describe your way of representing a rotation, and conversion to a rotation matrix. I think using matrices will help you, essentially these are a way of storing three linear combinations of three values.--Salix (talk): 08:03, 11 November 2012 (UTC)

Thanks, so I'll go (Euler axis/angle) -> (rotation matrix) -> (yaw, pitch, roll). I don't care for the matrix though. 105.236.57.198 (talk) 14:06, 11 November 2012 (UTC) Eon

November 11

Calculus notation

What type of function is a "d" with a forward-slanted strikethrough, I think it is a type of differencial.

An example: d(U - TS) less than or equal to (d̸)w_expansion + (d̸)w_nonexpansion. Plasmic Physics (talk) 00:36, 11 November 2012 (UTC)

In thermodynamics the state functions have perfect differentials (d), and then there are differentials d̸ which are not perfect, meaning that they are not differentials of state functions. For example, in a fluid the increase of energy dE = TdS−PdV where T is temperature, dS is increase in entropy, P is pressure, and dV is increase in volume. Here dE and dS and dV are perfect differentials, because E and S and V are state functions. For reversible processes TdS=d̸ Q is the heat transferred to the fluid, and PdV=d̸ W is the work done by the fluid. But 'heat content' Q and 'work content' W are not state functions. Bo Jacoby (talk) 01:28, 11 November 2012 (UTC).

Explicit solution that gives the n^th Harmonic number

Does one exist without integrals or other such operations that require the further determination of a solution? --Melab±1 ☎ 03:07, 11 November 2012 (UTC)

I'm fairly certain the harmonic number function is not elementary. As such there is no "explicit solution" for it in the form you seem to want. If you're satisfied with an approximation, taking a few terms from the asymptotic expansion works well. -- Meni Rosenfeld (talk) 06:08, 11 November 2012 (UTC)

What does f ⊑ g mean?

I recently read a paper that used the notation "f ⊑ g" without defining the symbol "⊑". It looks like a kind of square "subset" symbol. f and g were functions. Does this notation mean anything to anyone? --Doradus (talk) 22:54, 11 November 2012 (UTC)

Could they have been partial functions? It often means "extends".--80.109.106.49 (talk) 00:42, 12 November 2012 (UTC)

Unicode Character 'SQUARE IMAGE OF OR EQUAL TO' (U+2291) – though this probably does not help much. — Quondum 01:06, 12 November 2012 (UTC)

Thanks for the tip on partial functions. When I get a chance, I'll re-read that passage and see if that explains it. --Doradus (talk) 18:46, 12 November 2012 (UTC)

Of course it would have been boring to tell us the title of the paper or even its topic. That would have made the question much too easy. Looie496 (talk) 03:33, 12 November 2012 (UTC)

Of course it would be silly just to ask for the information you want. Much better to post a sarcastic remark insinuating that every newcomer asking a question ought to know the right amount of context to provide. --Doradus (talk) 18:46, 12 November 2012 (UTC)

To me it just means "f is a subset of g", but I could be wrong. Perhaps they are used in Cauchy spaces, but then again, perhaps not. Is there a way to search for such a symbol encyclopedia-wide in Wikipedia math-notation Special:Search function? ~AH1 ^(discuss!) 04:01, 13 November 2012 (UTC)

Actually Wikipedia search turns up zero hits for that symbol. Perhaps the Search function has some trouble with unusual unicode characters? --Doradus (talk) 13:38, 13 November 2012 (UTC)

For anyone who is interested, the paper is this one: "Strictness and Binding-Time Analyses: Two for the Price of One". It doesn't seem to be available for free online. The sentence is:

A domain projection γ on a domain D is a continuous function γ: D → D such that (i) γ ⊑ ID, and (ii) γ ∘ γ = γ (idempotence).

I take it that "ID" is the identity function over D. (perhaps it should have been I_D?) --Doradus (talk) 13:49, 13 November 2012 (UTC)

(edit conflict) This is the unicode character U+2291 (Square Image Of Or Equal To) (see for example this website). The German Wikipedia lists it in the article Unicodeblock Mathematische Operatoren. According to page 5 of this pdf file it denotes some kind of relation. but I don't know to what mathematical objects this is applied It seems Doradus found something. -- Toshio Yamaguchi (tlk−ctb) 13:52, 13 November 2012 (UTC)

Seeing the word “domain”: ⊑ is commonly used to denote partial orders (domains) in domain theory. If D, E are domains, the order on the function space D → E is defined pointwise, i.e., f ⊑ g iff f(x) ⊑ g(x) for every x ∈ D. So, if ID is indeed the identity function as you suggest, then (i) translates to γ(x) ⊑_D x for every x, where ⊑_D is the order supplied by the definition of D being a domain.—Emil J. 14:30, 13 November 2012 (UTC)

I had a look at the paper, and indeed it seems to be talking about domain theory (with some category theory mixed in). However, the author is very reader-unfriendly when it comes to providing context, or definitions of the concepts he’s using (let alone notation).—Emil J. 14:53, 13 November 2012 (UTC)

That sounds like the definition of a retract (in the sense used in topology, not in the sense used in category theory, see retract (category theory) for that). — Tobias Bergemann (talk) 14:32, 13 November 2012 (UTC)

Terrific! Thanks again, everyone, for all the leads. --Doradus (talk) 20:59, 17 November 2012 (UTC)

Fibonacci numbers

Hello, could anyone explain me Fibonacci sequence in short? And what is the meaning? Thanks. Bennielove (talk) 23:43, 11 November 2012 (UTC)

The article Fibonacci numbers starts at a fairly basic level. Please let us know what information there you would like more information on.Naraht (talk) 00:12, 12 November 2012 (UTC)

1 + 0 is 1, 1 + 1 is 2, 2 + 1 is 3, 3 + 2 is 5, 5 + 3 is 8, 8 + 5 is 13, 13 + 8 is 21, 21 + 13 is 34, and so on. Placing these numerical values into the side lengths of adjacent squares gives you the Fibonacci spiral, which is found everywhere from Ammonites to pinecones to the fractal-like interactions of tropical cyclones and just everything else found in nature. See Golden ratio and natural computing for some direct applications. ~AH1 ^(discuss!) 04:04, 13 November 2012 (UTC)

Ahem no. Please read Fibonacci Flim-Flam to find most of these statements to be debunked myths. "Nature has many spiral forms. None of them are golden spirals. Many don't even come close. None of them are 'explained' by Fibonacci mathematics." --KnightMove (talk) 12:32, 14 November 2012 (UTC)

November 12

Lagrangian density

Can someone explain to me Lagrangian density used to derived field equations like how ordinary lagrangian derives the trajectory of a particle? For example I don't understand what L is a function defined on for the density version, I know for ordinary Lagrangian L is defined on R^2n for n dimensional space, but I don't understand the field version. And also what's the motivation behind it? Ordinary Lagrangian wants the stationary point of the action due to a trajectory, so what about Lagrangian density? Money is tight (talk) 08:44, 12 November 2012 (UTC)

Perhaps the attached image will help. ~AH1 ^(discuss!) 04:06, 13 November 2012 (UTC)

This is a bit beyond me, but try looking at [1] - see Section 3 (p22) 'Lagrangian Mechanics' which leads into 3.2 'Principles of Lagrangian Field Theory' Christopherlumb (talk) 17:19, 14 November 2012 (UTC)

Lagrangian density is a function which integrated over spatial coordinates gives you the Lagrangian, as per the usual definition of density. Since you are dealing with fields, and not particles, in field theory, there is a contribution from each volume element to the total Lagrangian.

November 13

General algorithm for finding quotient rings.

Let $R$ be a ring and $(f_{1},...,f_{n})$ be the ideal generated by the elements $f_{1},...,f_{n}\in R[x]$. Is there a general algorithm for finding $R[x]/(f_{1},...,f_{n})$ up to isomorphism?--AnalysisAlgebra (talk) 05:27, 13 November 2012 (UTC)

November 14

2012x2015

Hello all. This is a puzzle that was in the newspaper (no prize or anything like that, don't worry); usually they are pretty easy but this one had me stumped. It went: Consider a 2012x2015 rectangle. Divide it into unit squares (1x1). How many of these unit squares does a diagonal of this rectangle pass through? I did some back of the envelope work with simpler squares (2x3, 3x5) and estimated that it had to be in the ballpark of 4030, but I couldn't find an exact answer (as 2012 and 2015 are coprime). I think I'm missing something really simple but my brain isn't working today. Help? 24.92.74.238 (talk) 03:49, 14 November 2012 (UTC)

Well, if it was exactly square, the answer would be the length of a side. Since it's slightly off square, I'd say it should be about double the longest length, or 4030. I assume that's how you got that answer. Not sure how to find the exact answer, though. StuRat (talk) 06:32, 14 November 2012 (UTC)

The answer is 4026. If you look at the line $y={\frac {2012}{2015}}x$, a segment of $x\in [m,m+1]$ will usually pass through 2 squares, but only one if for some integer n, $n\leq {\frac {2012}{2015}}x<{\frac {2012}{2015}}(x+1)\leq n+1$. This happens if $2012m\in \{0,1,2,3\}\ \mathrm {mod} \ 2015$ or equivalently $3m\in \{0,2012,2013,2014\}\ \mathrm {mod} \ 2015$. The solutions are $m\in \{0,671,1343,2014\}$. -- Meni Rosenfeld (talk) 06:35, 14 November 2012 (UTC)

I just ran a simulation and got 4025. You didn't count both (0,0) and (2012,2015) as squares, did you ? StuRat (talk) 07:07, 14 November 2012 (UTC)

No, I didn't. My result is also confirmed by 1Ouch0's solution, which is much more elegant (and general) than mine and easily verifiable for some simple special cases such as nXn and nX1.

Can you share your code? -- Meni Rosenfeld (talk) 07:45, 14 November 2012 (UTC)

Included below. StuRat (talk) 08:26, 14 November 2012 (UTC)

Simulation code (FORTRAN).

program DIAGONAL implicit none logical*1 TILE_OCCUPIED(0:2016,0:2016) integer*2 I,J,K,COUNT real*8 X,Y Initialization: do I = 0,2016 do J = 0,2016 TILE_OCCUPIED(I,J) = .false. enddo enddo COUNT = 0 Body: do K = 1,2015 X = 1.0*K - 0.00000001 Y = X*2012/2015 I = X ! Truncates. J = Y ! Truncates. if (.not. TILE_OCCUPIED(I,J) .and. (I .gt. 0) .and. (J .gt. 0)) then TILE_OCCUPIED(I,J) = .true. COUNT = COUNT + 1 print *,"Tile",COUNT,"= (",I,",",J,")" endif X = 1.0*K + 0.00000001 Y = X*2012/2015 I = X ! Truncates. J = Y ! Truncates. if (.not. TILE_OCCUPIED(I,J) .and. (I .gt. 0) .and. (J .gt. 0)) then TILE_OCCUPIED(I,J) = .true. COUNT = COUNT + 1 print *,"Tile",COUNT,"= (",I,",",J,")" endif enddo Termination: print * print *,"Count = ",COUNT end

Simulation solution set.

Tile 1 = ( 1 , 1 ) Tile 2 = ( 2 , 1 ) Tile 3 = ( 2 , 2 ) Tile 4 = ( 3 , 2 ) Tile 5 = ( 3 , 3 ) Tile 6 = ( 4 , 3 ) Tile 7 = ( 4 , 4 ) Tile 8 = ( 5 , 4 ) Tile 9 = ( 5 , 5 ) Tile 10 = ( 6 , 5 ) Tile 11 = ( 6 , 6 ) . . . (See the rest here: [2].) . . . Tile 4016 = ( 2010 , 2007 ) Tile 4017 = ( 2010 , 2008 ) Tile 4018 = ( 2011 , 2008 ) Tile 4019 = ( 2011 , 2009 ) Tile 4020 = ( 2012 , 2009 ) Tile 4021 = ( 2012 , 2010 ) Tile 4022 = ( 2013 , 2010 ) Tile 4023 = ( 2013 , 2011 ) Tile 4024 = ( 2014 , 2011 ) Tile 4025 = ( 2015 , 2012 ) Count = 4025

Your solution set is missing (2014,2012). Which is a result of using an incorrect offset for the diagonal. In your calculation x=2 corresponds to the right edge of the leftmost square, which means the corresponding y-value is 1 + 2012/2015, and not 2*2012/2015 as in your code. By the time it gets to x=2015-ε, it is thinking it is exiting (2014,2011) through a corner.

It's better to think of leftmost square as [0,1]X[0,1], and since you round down the real values, it should be denoted (0,0). So you should let K run from 0 to 2015, and instead of $I>0,\ J>0$ require that $0\leq I\leq 2014,\ 0\leq J\leq 2011$. -- Meni Rosenfeld (talk) 12:51, 14 November 2012 (UTC)

The part which confused me is that my values seemed to match your statement that "The solutions are $m\in \{0,671,1343,2014\}$". That is, I only get one tile on each of those columns, including the 2014 column. If I shift to allow tiles (0,0) and (0,1), and exclude tile (2015,2012), then I do get 4026 tiles, but, oddly, there are now two tiles in column 2014, at (2014,2011) and (2014,2012). StuRat (talk) 19:47, 14 November 2012 (UTC)

If (0,0) is the bottom-left square, then (0,1) shouldn't be in the solution set, so something is still off. I think you shifted by decreasing both X and Y by 1 (which gives you the same wrong offset), where what you should have done is decrease X by 1 and calculate Y based on the new X (meaning it decreases by 2012/2015). -- Meni Rosenfeld (talk) 22:03, 14 November 2012 (UTC)

(ec) Sturat and Meni beat me to it.

You were almost there...

X and Y are coprime, that means that the diagonal will only enter one square at a vertex, it will enter (Y - 1) squares via the bottom line, and (X - 1) squares via the left - so the answer is X + Y - 1, whether you are looking at 2x3, 3x5, or 2012x2015.

If X and Y are not coprime, subdivide the problem into n rectangles of size p times q, where p and q are coprime. So it'll be n (p + q - 1) which reduces to X + Y - n, n being the gdc of X and Y.

Cheers. - ¡Ouch! (^{hurt me} / _{more pain}) 06:59, 14 November 2012 (UTC)

Another way to express essentially the same reasoning (coprime case only): your diagonal has to cross X-1 vertical lines and Y-1 horizontal lines (excluding the borders of the grid). If X and Y are coprime the diagonal never passes through an interior vertex of the grid, that is it doesn't cross a vertical line and a horizontal line at the same time. Hence the diagonal intersects grid lines at X+Y-2 distinct points, and is thus divided in X+Y-1 distinct intervals. That is, it crosses exactly X+Y-1 unit squares of the grid. --FvdP (talk) 12:21, 16 November 2012 (UTC)

November 15

Logic puzzle questions

These are questions rather for the "humanities" or "miscellaneous" desk, but I have a feeling that my chances to get an answer are best here.

1. Who invented the logic grid puzzle, and when? Was the Zebra puzzle, brought to a wider public by Life International in 1962, maybe the first one of its kind?
2. The text published in there does not mention any involvement of Einstein, which implies that this "information" was added later. However, it could be possible that it was mentioned in the removed images. Has anybody seen the original and can tell for sure?

--KnightMove (talk) 07:53, 15 November 2012 (UTC)

Algebra questions of a certain type.

Find a surjective group homomorphism $S_{5}\rightarrow S_{4}$ or prove that one does not exist. What is the strategy in general for finding surjective homomorphisms?--AnalysisAlgebra (talk) 09:43, 15 November 2012 (UTC)

Say we wish to map a group G onto a group H. If we know a generating set for H, clearly we want to make sure that it is contained in the image of our map - then we're done. This can be easier said than done, however! In the case of finite groups, you can search for normal subgroups of size |G|/|H|, since the kernel of your proposed map will have to be such a thing. For the symmetric group, this is not *such* a Herculean task, because we know that conjugation by the symmetric group acts transitively on the set of elements of a fixed cycle-type. (So, if the normal subgroup contains a 3-cycle, it actually contains *all* 3-cycles, etc.) Our article symmetric group discusses just what normal subgroups a symmetric group can have. Sorry if this answer was too specific to your one example - it can be quite ad hoc at times, finding epimorphisms. Icthyos (talk) 10:45, 15 November 2012 (UTC)

So the kernel has to be a normal subgroup of size 5. So the answer is that it's not possible, since no subgroup of $S_{5}$ has 5 elements. This is because there is no way to sum the orders of selected conjugacy classes to 5.--AnalysisAlgebra (talk) 18:59, 15 November 2012 (UTC)

Yup. There's a stronger fact that's worth mentioning: apart from a few (two/three?) exceptions, each symmetric group has only one proper, non-trivial normal subgroup: its alternating group. (And we know what size it has.) Icthyos (talk) 09:54, 16 November 2012 (UTC)

(Just to be precise: $S_{5}$ does have subgroups of order five, but no normal subgroups of order 5.) Icthyos (talk) 12:30, 16 November 2012 (UTC)

Statistical analysis of exam passes

Hi, I have some data about some students who have to take a series of 13 exams and are expected to pass a certain number after a set number of 6 month periods. So I have the following table of "expected" passes each 6 month period:

Exam session                   1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 
                               =============================================
Number of expected exam passes 0 1 3 5 7 8 8 9 10 11 12 13 

Then I have the dates of the month that each student passed each exam (which can easily be converted to total number of exams passed at each 6 month period for each student). Eg:

Exam Period 1 2 3 4 5 6 7  8  9 10 11 12 13 14 15 16 17 
            ===========================================
Passes      0 0 0 1 3 5 7 10 10 10 11 11 12 12 13 13 13 

(This student clearly not passing fast enough compared to the above table)

What statistical technique would be suitible to analyse whether the above table is reasonable compared to the actual performance of the student, that is, is the expected numbers too harsh, or too lenient given actual students performance?

I am just looking for the name of a relevant technique, wikipedia should suffice for the rest as I am a fairly competent mathematician (just not statistician).

Thanks for any help in advance,

Matt 80.254.147.164 (talk) 11:08, 15 November 2012 (UTC)

Maybe I'm misunderstanding your set-up. It seems to me that as of the end of each of the 12 six-month periods for which data are given, the student has equaled or exceeded (actually exceeded, until right near the end) expectations. Why do you say This student clearly not passing fast enough compared to the above table? Duoduoduo (talk) 15:50, 15 November 2012 (UTC)

Sorry, I copied it from another source and didnt have zeroes in the right places, edited and fixed.80.254.147.164 (talk) 16:22, 15 November 2012 (UTC)

I edited your post to line up the columns, although I had to make a guess on the first part, since the number of columns don't match. Are you saying all 13 exams should be satisfactorily completed by the 12th period ? If so, I suggest you just list the number of periods behind or ahead for each student, and then find the difference in this number from period to period. In this case:

Exam Period  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 
             ====================================================
Passes       0  0  0  1  3  5  7 10 10 10 11 11 12 12 13 13 13 13
Expectation  0  1  3  5  7  8  8  9 10 11 12 13 13 13 13 13 13 13
             ====================================================
Comparison   0 -1 -3 -4 -4 -3 -1 +1  0 -1 -1 -2 -1 -1  0  0  0  0
Difference    -1 -2 -1  0 +1 +2 +2 -1 -1  0 -1 +1  0 +1  0  0  0

You could then total all the differences in the comparisons for all the students. This suggests that you need to give more time where the differences are negative, and less where they are positive. However, this doesn't directly account for knowledge that some exams might be harder than others. To do this, you might want to break it down by exam:

Exam                             1    2    3    4    5    6    7    8    9   10   11   12   13
                               ===============================================================
Periods allocated to pass exam   2  1/2  1/2  1/2  1/2  1/2  1/2    1    0    1    1    1    1
Periods taken to pass exam       4  1/2  1/2  1/2  1/2  1/2  1/2  1/3  1/3  1/3    3    2    2
                               ===============================================================
Difference                      -2    0    0    0    0    0    0 +2/3 -1/3 +2/3   -2   -1   -1

This suggests you need to allocate more time for the first exam and last 3 (of course, this assumes the student you listed is typical, you need to total your results to see the real patterns). StuRat (talk) 16:44, 15 November 2012 (UTC)

Property of the parabola

I'm looking for a textbook (or website) that mentions a property of the parabola. If it exists, it is likely to be a textbook on analytic geometry or an introductory textbook on differential calculus. I can describe this property using a theorem. It might say something like the following.

• If (x_A, y_A) and (x_B, y_B) are the coordinates of two points, A and B, on a parabola, the line joining these two points is parallel to the tangent at point C where the x coordinate of C is:

$x_{C}={\frac {x_{A}+x_{B}}{2}}$

The value of this theorem would be that it enables students who are about to be introduced to differential calculus to calculate the slope of a tangent to a parabola (ie the derivative of a quadratic function) without requiring an understanding of the limit or the derivative. If asked to find the slope of the tangent to a given parabola at the point where x=x₁ students can select any two values of x equidistant from x₁, find the values of the function for those two values of x, and then find the slope of the line joining the two points.

slope: $m={\frac {f(x_{1}+\Delta )-f(x_{1}-\Delta )}{2\Delta }}$

This theorem is reminiscent of the mean value theorem but of course the MV theorem is unnecessarily advanced for students about to be introduced to differential calculus.

The only proof I can offer is one based on differential calculus - the following equation is true for any second degree polynomial but not for any higher-degree polynomial. (It is also trivially true for linear functions and constants.)

${\frac {f(x_{B})-f(x_{A})}{x_{B}-x_{A}}}=f^{\prime }\left({\frac {x_{A}+x_{B}}{2}}\right)$

Any suggestions about a textbook? Dolphin (t) 14:50, 15 November 2012 (UTC)

A non-calculus proof goes like this: use your expression for the arc slope m , plug in $f(x)=x^{2}+ax+b$ for both $x+\Delta$ and $x-\Delta$, and see that the result $m=2x+a$ is independent of $\Delta$ (for $\Delta$ not equal to zero, but arbitrarily small). That provides the intuition. Is that what you wanted? Duoduoduo (talk) 16:24, 15 November 2012 (UTC)

Maybe this passage from Polynomial long division will help:

Polynomial long division can be used to find the equation of the line that is tangent to a polynomial at a particular point.[Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", Mathematical Gazette 89, November 2005: 466-467.] If R(x) is the remainder when P(x) is divided by (x – r )² — that is, by x² – 2rx + r ² — then the equation of the tangent line to P(x) at x = r is y = R(x) (regardless of whether or not r is a root of the polynomial).

This gives a non-calculus way to get the tangent's slope, although I don't know if that method can be made intuitive for a student who doesn't know calculus. Duoduoduo (talk) 16:34, 15 November 2012 (UTC)

Another way, using this Theorem: Lets have a family of parallel lines which intersect the parabola. On each line we mark the middle point between the two points of intersection of the line with the parabola. Then all the points we marked are on a single vertical line. --84.229.146.110 (talk) 18:35, 15 November 2012 (UTC)

Transitive actions

Does there exist a transitive action of $S_{5}$ on a set with 7 elements?--AnalysisAlgebra (talk) 15:21, 15 November 2012 (UTC)

Try using the orbit-stabiliser theorem. It's very helpful for this sort of problem, especially since you require the action to be transitive, so there is only one orbit. Icthyos (talk) 15:30, 15 November 2012 (UTC)

So again the answer is no, since 7 does not divide 120.--AnalysisAlgebra (talk) 19:00, 15 November 2012 (UTC)

Rings.

Let $R$ be a ring and $I,J$ ideals. Show that $R=I+J\implies I\cup J=IJ$

I can derive that $IJ\subset I\cup J$ (it is true in general without the given condition, in fact) but that's all.--AnalysisAlgebra (talk) 19:03, 15 November 2012 (UTC)

That's because it isn't true. Consider $R=\mathbb {Z}$ , $I=5\mathbb {Z}$ and $J=7\mathbb {Z}$, say. A typo in the statement, probably because someone was standing on their head.John Z (talk) 21:50, 15 November 2012 (UTC)

Yes, that is a typo. The statement is actually $R=I+J\implies I\cap J=IJ$. That's what you get for standing on your head. Anyway $IJ\subset I\cap J$ as well.--AnalysisAlgebra (talk) 03:58, 16 November 2012 (UTC)

Well, the other way is pretty easy too. Consider what R = I+J means for explicit elements of R, I & J, by the definition of the sum of two ideals. And then use that to get the other inclusion.John Z (talk) 05:53, 16 November 2012 (UTC)

Is any conditional probability a Bayesian probability?

Are both synonym? Comploose (talk) 19:06, 15 November 2012 (UTC)

No, the terms are orthogonal. A conditional probability can be interpreted in either a frequentist or Bayesian way. If X is the result of throwing a fair die, then $\mathrm {Pr} (X=1|X<4)={\tfrac {1}{3}}$. The frequentist interpretation of this is that if you toss a die many times, and consider only those time which resulted in less than 4, about a third of those will result in 1. The Bayesian interpretation is that if I toss one die, and observe that the result is less than 4, my subjective probability for it being 1 is updated to 1/3.

Conversely, if you are a Bayesian agent who has not yet observed any evidence and is working by your prior, your Bayesian subjective probability will not be a conditional probability. -- Meni Rosenfeld (talk) 19:34, 15 November 2012 (UTC)

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)

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)