Wikipedia:Reference desk/Archives/Mathematics/2010 July 6

= July 6 =

6 sided die different from the standard
So I hear there is a way to rearange the numbers on a pair of 6 sided die which are different from the standard die and which have the same probabilities and outcomes when you toll both and add the outcome.... What are the numbers on each face of the 2 dice? —Preceding unsigned comment added by KatRosenbaum (talk • contribs) 04:34, 6 July 2010 (UTC)
 * If you're required to use the same collection of numbers (1,1,2,2,3,3,4,4,5,5,6,6) then I think it's impossible. Getting the right probability for 2 and 12 (1/36 each) requires the 1s and 6s to be on opposite dice.  Then for getting the right probability for 4 (3/36), the chance of getting a 1 and 3 combo is always 2/36 since where ever each 3 is, there's exactly one 1 on the opposite die.  To get the final combination that leads to a 4 requires the 2s to be on opposite dice.  Similarly the 5s must be on opposite dice.  Then to get the right probability for 6 (5/36) the 3s must be on opposite dice, and similarly for the 4s.
 * If you can use a different collection of numbers then it would be pretty easy. An obvious way would be to put 0,1,2,3,4,5 on one die and 2,3,4,5,6,7 on the other. Rckrone (talk) 06:31, 6 July 2010 (UTC)


 * See Sicherman dice. --Matthew Auger (talk) 07:42, 6 July 2010 (UTC)

There was an article in the Monthly a few years ago titled Dice With Fair Sums. Try that on Google Scholar. Michael Hardy (talk) 20:47, 6 July 2010 (UTC)
 * ...April 1988, pages 316 through 328. Michael Hardy (talk) 20:49, 6 July 2010 (UTC)

Equating an exponential function to a linear function
$$e^y=(e-1)y+1$$

Solve for y

How do you solve an equation wherein an exponential function is equated to a linear function ?––220.253.104.38 (talk) 07:10, 6 July 2010 (UTC)
 * There's not an elementary way to solve for y from such an equation, however in this particular case there are some obvious values of y that should jump out that you might guess and check. Rckrone (talk) 07:31, 6 July 2010 (UTC)
 * Ah, yes, y=0 and y=1 are solutions. Are there any others?––220.253.104.38 (talk) 07:33, 6 July 2010 (UTC)
 * You might want to try a bit of graph sketching. -- The Anome (talk) 07:55, 6 July 2010 (UTC)
 * And convex functions, also. --78.13.141.245 (talk) 07:37, 7 July 2010 (UTC)

Sum of exponentials
$$2^x + 2^{1-x}-3=0$$

solve for x

How do you solve an equation involving the sum of multiple exponentials?––220.253.104.38 (talk) 07:54, 6 July 2010 (UTC)
 * This looks very similar to the problem above. This leads me to think this might be a homework assignment: please read the comments at the top of this talk page. -- The Anome (talk) 07:57, 6 July 2010 (UTC)
 * No, it doesn't look similar to the one above at all. It's much simpler than the one above. Michael Hardy (talk) 17:55, 6 July 2010 (UTC)
 * This is a different type of question to the one bove and can be solved in a couple of ways. For starters try thinking of 2x as a unit rather than x itself. Forget to ask, first of all, have you come across complex numbers? Dmcq (talk) 08:03, 6 July 2010 (UTC)
 * And if you haven't come across complex numbers, or perhaps even if you have, are you certain about the signs in the equation? Dmcq (talk) 08:09, 6 July 2010 (UTC)
 * should be –3, not +3, oops.––220.253.104.38 (talk) 08:15, 6 July 2010 (UTC)
 * $$2^x + 2(2^x)^{-1}-3=0$$--220.253.104.38 (talk) 08:17, 6 July 2010 (UTC)
 * Actually the thinking of the previous section might be more use to you now you have that minus sign in there. Otherwise if you want to go along this line doing something like z=2x is how the chunking is normally done. Dmcq (talk) 08:28, 6 July 2010 (UTC)
 * Guess and check is so dodgy, I much prefer to solve it algebraically. Anyway, I've solved it now, thanks for your help.––220.253.104.38 (talk) 08:33, 6 July 2010 (UTC)

Substitution: Let
 * $$ u = 2^x \, $$

Then
 * $$ \frac{1}{u} = 2^{-x} $$

So
 * $$ 2^x + 2\cdot 2^{-x} - 3 = 0 $$

becomes
 * $$ u + 2\frac{1}{u} - 3 = 0 $$

Multiply both sides by u:
 * $$ u^2 + 2 - 3u = 0 \, $$
 * $$ u^2 + 2 - 3u = 0 \, $$

That's an ordinary quadratic equation. Once you've solved it for u, you've got the value of 2x. Take its base-2 logarithm to get the value of x. Michael Hardy (talk) 17:54, 6 July 2010 (UTC)

Connections and Differential Forms
Let $$\scriptstyle M$$ be an $$\scriptstyle n$$-dimensional, real differentiable manifold and let $$\scriptstyle \nabla$$ be a smooth connection on $$\scriptstyle M$$. So, for a pair of vector fields on $$\scriptstyle M$$, say $$\scriptstyle X$$ and $$\scriptstyle Y$$, we have a new vector field $$\scriptstyle \nabla_XY$$. Now let $$\scriptstyle \omega$$ be a differential n-form on $$\scriptstyle M$$. I've never quite understood how to understand the expression $$\scriptstyle \nabla\omega$$. I've seen expressions like
 * $$ \scriptstyle \nabla_X [ \omega (Y_1,\ldots,Y_n) ] \, = \, [\nabla_X\omega](Y_1,\ldots,Y_n) \, + \, \omega(\nabla_XY_1,Y_2,\ldots,Y_n) \, + \, \omega(Y_1,\nabla_XY_2,Y_3,\ldots,Y_n) \, + \, \cdots \, + \, \omega(Y_1,\ldots,Y_{n-1},\nabla_XY_n).$$


 * $$\scriptstyle \iff [\nabla_X\omega](Y_1,\ldots,Y_n) \, = \, \nabla_X [ \omega (Y_1,\ldots,Y_n) ] \, - \, \omega(\nabla_XY_1,Y_2,\ldots,Y_n) \, - \, \omega(Y_1,\nabla_XY_2,Y_3,\ldots,Y_n) \, - \, \cdots \, - \, \omega(Y_1,\ldots,Y_{n-1},\nabla_XY_n).$$

And this makes sense from a formal point of view. But thinking about it; it doesn't make any real sense to me. For example; $$\scriptstyle \omega (Y_1,\ldots,Y_n)$$ is a differential form, and not a vector field. But $$\scriptstyle \nabla$$ is a connection, i.e. it takes a pair of vector fields and gives another vector field. (It's a type of covariant derivative.) Out of the last displayed line, the only term I don't understand is $$\scriptstyle \nabla_X [ \omega (Y_1,\ldots,Y_n) ]$$. How can I make sense of it? •• Fly by Night (talk) 20:26, 6 July 2010 (UTC)


 * In $$\scriptstyle \nabla_X [ \omega (Y_1,\ldots,Y_n) ]$$, $$\scriptstyle\omega (Y_1,\ldots,Y_n)$$ is a function on $$\scriptstyle M$$. The whole expression then means the vector field $$\scriptstyle X$$ applied to this function. Intuitively $$\scriptstyle [\nabla_X \omega]$$ is defined so that something like the product rule holds (ie. the first formula in your if and only if holds).Holmansf (talk) 00:20, 13 July 2010 (UTC)

Using the moment generating function to find E(y)
The p.d.f. is $$f(y) = \frac{1}{2}e^{\left|y\right|}, -\infty < y < \infty$$. When I find the moment generating functions, I set the two cases where one is when y is positive and the other where y is negative. $$m(t) = -\frac{1}{2(t-1)} $$ When y is negative, and t < 1 for the integral to converge. $$m(t) = \frac{1}{2(t+1)} $$ When y is positive, and t > 1 for the integral to converge. Now, since E(y) is found by setting t = 0, I use the first m.g.f. to calculate the E(y) since that is when t < 1. I get the answer of -1/2 for E(Y).

However, when I try to integrate y * p.d.f. over the ranges from negative infinity to infinity, I get 0. I'm wondering where did I go wrong here. 142.244.151.247 (talk) 23:17, 6 July 2010 (UTC)
 * The moment generating function is an integral over y, not a function of y. There are not two cases. Bo Jacoby (talk) 06:14, 7 July 2010 (UTC).
 * The integral of ½eundefined from –∞ to ∞ is infinite, whereas the integral of a pdf over its support should be 1. Qwfp (talk) 07:37, 7 July 2010 (UTC)
 * The OP probably ment to say ½e&minus; Bo Jacoby (talk) 07:55, 7 July 2010 (UTC).
 * Yeah, I meant to say $$f(y) = \frac{1}{2}e^{-\left|y\right|}, -\infty < y < \infty$$, sorry about that. What do you exactly mean by integral over y, not function of y? 142.244.148.125 (talk) 14:24, 7 July 2010 (UTC)
 * In that case your mgf is wrong, and once you have the right one you need to differentiate it with respect to t before setting t to zero to find E(y), as 'moment generating function' explains. Qwfp (talk) 14:49, 7 July 2010 (UTC)
 * I know the mgf is wrong, but I'm wondering shouldn't there be two cases for mgf when y is positive and when y is negative here, since there's an absolute value involved here. 142.244.148.125 (talk) 14:58, 7 July 2010 (UTC)
 * The mgf m(t) is for any given t defined by $$\int_{-\infty}^{+\infty}\frac12e^{-|y|}e^{ty}\,dy$$ which does not depend on y (that's a bound variable), only on t, so there cannot be any cases depending on whether y is positive or negative. When you evaluate the integral properly, you will actually end up with three cases for t ≤ −1, −1 < t < 1, and t ≥ 1.—Emil J. 14:59, 7 July 2010 (UTC)