Wikipedia:Reference desk/Archives/Mathematics/2013 April 19

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April 19

Factorial and self-power

Are f(x) = x^x and g(x) = x! Π(x) related by transformations? They appear rather similar to me as shown in the image (image removed for space) Plots of x^x and x!. . Pokajanje|Talk 01:06, 19 April 2013 (UTC)

They are related by Stirling's approximation: $x!\approx x^{x}e^{-x}{\sqrt {2\pi x}}$. Looie496 (talk) 01:45, 19 April 2013 (UTC)

Note, however, that f(x) is a continuous function for positive reals x but g(x) is only defined for non-negative integers. Extensions of n! to non-integer values is possible, but your image of y = g(x) should really show only discrete points (0, 1), (1, 1), (2, 2), (3, 6), (4, 24), etc. EdChem (talk) 07:22, 19 April 2013 (UTC)

Blame GeoGebra. Pokajanje|Talk 15:06, 19 April 2013 (UTC)

(ec) Or, alternatively, keep the curve and change the label to Π(x) or Γ(x+1). Double sharp (talk) 15:12, 19 April 2013 (UTC)

Continuing on the topic of x^x, why is it that graphing software plots no values for x < 0? The rule is that x^x ∉ ℝ only if x < 0 and $x={\frac {2k+1}{2n}};k,n\in \mathbb {Z}$ Pokajanje|Talk 02:48, 24 April 2013 (UTC)

I guess that is computed by x^x = e^{x log(x)}, and the logarithm is imaginary for negative reals. Bo Jacoby (talk) 09:31, 24 April 2013 (UTC).

Testing a few values, I'm guessing the graph below 0 would look like a sinusoidal wave that gradually became horizontally compressed (e.g. -1^-1 = -1, $(-{\frac {2}{3}})^{(}{\frac {-2}{3}})={\frac {3}{2}}^{\frac {2}{3}}\approx 1.31037$).

Minimising

If I want to optimise a process so that two (positive) observed numbers, say a and b, are both as small as possible, does it make most sense to minimise a^2 + b^2? — Preceding unsigned comment added by 86.171.43.198 (talk) 23:27, 19 April 2013 (UTC)

We'd need a lot more info. For example, if "a" varies from 1-2, and "b" varies from 1000-2000, then your method would basically just minimize "b" and ignore "a". Also, are "a" and "b" equally important, or is minimizing one more important than the other ? Let me come up with a formula, if we say that "b" is "n" times as import to minimize (we also need to know the minimum and maximum values of each):

[a/(a_min+a_max)] + [n*b/(b_min + b_max)]

If you don't know the minimum and maximum values, but do know the averages, you can substitute those in, instead. (You will note that I didn't divide by 2 when figuring the average of the min and max, since it "cancels out" when doing minimization).

Of course, this is just one way to do it. You might decide that "b" can absolutely never be allowed to go over "x", for example, which would then require a tweak to the formula.

Ideally, you would be able to combine "a" and "b" into a single factor "c", and then minimize for it. For example, if "a" is miles driven by delivery trucks, and "b" is amount of overtime paid, you could determine how much each of those cost your company, and combine both in a formula to get the total cost "c", as a function of "a" and b", then minimize for that. But, if there's no way to combine them together like this, then my previous answer is the best you can do. StuRat (talk) 23:33, 19 April 2013 (UTC)

They are equally important and of equal expected magnitude / range of magnitudes. 86.171.43.198 (talk) 00:25, 20 April 2013 (UTC)

Have a look at Multi-objective optimization there are many different techniques including No-preference methods which includes minimising $|a|+|b|$ (a $L^{1}$ norm), $a^{2}+b^{2}$ (a $L^{2}$ norm) or $\max(|a|,|b|)$ (a $L^{\infty }$ norm). --Salix (talk): 03:25, 20 April 2013 (UTC)