Wikipedia:Reference desk/Archives/Mathematics/2008 August 8

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August 8

Gaussian function

I'm trying to prove to myself something at the Gaussian function article. The article says that the Gaussian integral

$\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}$

and that therefore

$\int _{-\infty }^{\infty }ae^{-{(x-b)^{2} \over 2c^{2}}}\,dx=ac\cdot {\sqrt {2\pi }}$

I understand why the a showed up in the answer: because it's a constant that was inserted after the integral sign. But I'm not sure how you would arrive at the second equation given the first equivalency (in other words, how you would express the left side of second equation in terms of the left side of the first). Could someone lead me through it? (I'm the family willy member of a scientist who wants to understand this for the purpose of their research.) Thanks! —anon —Preceding unsigned comment added by 70.23.85.94 (talk) 02:44, 8 August 2008 (UTC)

You need to use integration by substitution. Specifically, set $u={\frac {x-b}{c{\sqrt {2}}}}$. Algebraist 02:51, 8 August 2008 (UTC)

By the way, from a geometrical point of view, this is completely obvious. Regarding the first integral as the area under the graph of $e^{-x^{2}}$, we obtain the second by shifting the graph horizontally a distance b (which doesn't change the area), stretching it vertically by a factor a (which multiplies the area by a), and the stretching it horizontally by a factor $c{\sqrt {2}}$ (which multiplies the area by $c{\sqrt {2}}$). Algebraist 03:09, 8 August 2008 (UTC)

Thanks so much for the u-substitution; I understand perfectly now. And thanks for the geometrical insight as well. —Preceding unsigned comment added by 70.23.85.94 (talk) 03:27, 8 August 2008 (UTC)

Sudoku minimum and maximum for fixed outcome

I was wondering what is a minimum number of digits that must be filled in in a sudoku grid before there is only one solution and, considering that it varies depending on where they are placed, what is the maximum number that can be filled in and still leave the puzzle with more than one solution. I was trying to figure it out on paper but got nowhere, and don't have the math to do it except by trial and error.--70.107.9.159 (talk) 04:25, 8 August 2008 (UTC)

The most that can be given without specifying the solution is all but four. Your other question is massively harder, and the answer is not known. The least anyone's come up with is 17 (in thousands of different ways). See mathematics of Sudoku for more. Algebraist 04:29, 8 August 2008 (UTC)

Also I think the majority of mathematicians who seriously studied this tend to believe this is actually the minimum, and call it the 17 conjecture. Of course that’s not everybody, but it seems to be the trend. There’s also a proven lower bound, but I can’t recall what it was – but I think something absurd like 6. GromXXVII (talk) 11:45, 8 August 2008 (UTC)

Thanks to you both. The linked page is very informative, if much of it is over my head. I was not aware there was so much already written on the subject (I naively even thought I might be raising a somewhat novel question).--70.107.9.159 (talk) 12:40, 8 August 2008 (UTC)

I don't think anything less than 8 numbers can be enough because then you can just swap two numbers everywhere. – b_jonas 11:12, 9 August 2008 (UTC)

That's probably the trivial lower bound GromXXVII was trying to remember. Algebraist 15:11, 10 August 2008 (UTC)

The Dildogarithm

I am trying to evaluate
$\operatorname {Li} _{2}(-1)=\int _{-1}^{0}{\frac {\ln(1-t)}{t}}dt=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k^{2}}}=-{\frac {\pi ^{2}}{12}}$
whose exact value is already well known. So, I picked the integral and tried so solve it by making the substitution u=1-t, du=-dt and transforming the integral to
$\int _{-1}^{0}{\frac {\ln(1-t)}{t}}dt=\int _{1}^{2}{\frac {\ln(u)}{1-u}}du=\int _{1}^{2}\ln(u)-{\frac {\ln(u)}{u}}du=\left[u\ln(u)-u-{\frac {(\ln(u))^{2}}{2}}\right]_{1}^{2}$
which comes out to an exact value of $\ln(4)-1-{\frac {(\ln(2))^{2}}{2}}$. What am I doing wrong?--A Real Kaiser...NOT! (talk) 05:55, 8 August 2008 (UTC)

$\int _{1}^{2}{\frac {\ln(u)}{1-u}}du=\int _{1}^{2}\ln(u)-{\frac {\ln(u)}{u}}du$ is bogus. ${\frac {a}{b+c}}\neq {\frac {a}{b}}+{\frac {a}{c}}$ --tcsetattr (talk / contribs) 06:06, 8 August 2008 (UTC)

UGGGGGHHHHHHHHHHHHHHH, I am so blind. I can't believe that I made an algebra one mistake.--A Real Kaiser...NOT! (talk) 07:28, 8 August 2008 (UTC)

Cosmological constant influencing the moon and pioneer orbits?

The article on the Kepler_problem_in_general_relativity assumes that the cosmological constant is zero. What happens to the field and to the orbit if this is not the case? Is there a connection to the pioneer anomaly? Bo Jacoby (talk) 07:53, 8 August 2008 (UTC).

Probably a question better for the science reference desk. GromXXVII (talk) 11:47, 8 August 2008 (UTC)

The observed cosmological constant is too small to matter. In the Newtonian approximation the cosmological constant behaves like a repulsive acceleration that's proportional to distance. The magnitude of the acceleration is Ω_v H_o² ≈ 10⁻²⁴ m/s² per A.U., which is a dozen orders of magnitude too small to explain the Pioneer anomaly, and has the wrong sign anyway. It's also too small to have a noticeable effect on planetary orbits. The effect gets larger with distance from the Sun, but even at the distance of the Oort cloud it only decreases the Sun's acceleration by about one part in a hundred million. -- BenRG (talk) 14:12, 8 August 2008 (UTC)

Thank you very much for the answer. Is this important information found anywhere in wikipedia? If no, shouldn't it? Bo Jacoby (talk) 20:41, 8 August 2008 (UTC).

I added a brief mention to Pioneer anomaly. Adding a cosmological constant to Kepler problem in general relativity would be a lot more work (not to imply it wouldn't be worthwhile). I wouldn't mind seeing the cosmological constant described somewhere in Newtonian terms, but it would probably need a reference, and I don't think I've actually seen a derivation of a Newtonian-approximation-with-cosmological-constant anywhere in print. -- BenRG (talk) 21:53, 8 August 2008 (UTC)

Modulus integrals

If f(x) is a function whose sign changes at least once in the interval a≤x≤b, then consider $\int _{b}^{a}f(x)\ dx$ and $\int _{b}^{a}|f(x)|\ dx$. Since the second of these is the first but with the negative regions above the x-axis, these two integrals have the same value as they bound the same area.

My question is, what is the difference between integrating these two integrals; are you just meant to use modulus signs when you integrate a function whose sign changes in the interval you're integrating over? 92.4.189.176 (talk) 10:50, 8 August 2008 (UTC)

I suspect you are thinking that $\int _{b}^{a}f(x)\ dx$ is the area enclosed between f(x) and the x axis. In fact, it is the signed area - areas below the x axis are subtracted - the introduction to our integral article explains this. But $\int _{b}^{a}|f(x)|\ dx$ does give you the (unsigned) area because |f(x)| is never negative. So the difference between $\int _{b}^{a}f(x)\ dx$ and $\int _{b}^{a}|f(x)|\ dx$ is that the first gives you the signed area and the second the unsigned area - and they will have different values if f(x) is negative anywhere between a and b. For example:

$\int _{-1}^{1}x\ dx=0{\mbox{ but }}\int _{-1}^{1}|x|\ dx=1$

because the areas to the left and right of the y axis are equal in magnitude (both 0.5) but have opposite signs. Gandalf61 (talk) 11:12, 8 August 2008 (UTC)

If f(x) changes sign n times, am I right in saying that you need to split it into n+1 integrals? 92.4.189.176 (talk) 11:37, 10 August 2008 (UTC)

That's right, although of course a sign change could occur at the end of the interval and the resulting 0-length interval would not have any effect. --Tardis (talk) 14:47, 11 August 2008 (UTC)

Unsolved Mathmatical Proof

I remember reading in one article that an organization was giving away something like 1 million dollars to whoever could prove some old mathematical concept. I remember them saying it was key to tons of work that had already been done. Does anyone know what it was. Thank you

Teopb (talk) 18:09, 8 August 2008 (UTC)

Probably thinking of Riemann hypothesis. Baccyak4H (Yak!) 18:19, 8 August 2008 (UTC)

Actually there are several. But Riemann is probably the most noteworthy, due to the fact that many, many proofs have been done which assume it true. Baccyak4H (Yak!) 18:24, 8 August 2008 (UTC)

I remember hearing about one where the guy with the money wanted a method for stacking boxes of (possibibly unrelated) different sizes to get the smallest load (ie fit on a truck of a certain size) - it may have been a simplification of another problem but the way it was put to me suggested that that was the problem itself...87.102.45.156 (talk) 21:33, 10 August 2008 (UTC)

That sounds like a description of the bin packing problem, an efficient solution to which would resolve the P = NP problem, one of the Millennium Prize Problems Baccyak4H referred to above. Alternatively, an efficient algorithm to solve Minesweeper (or a proof that no efficient solution exists) would do it. Algebraist 21:43, 10 August 2008 (UTC)

"Three-edge Coloring"

In the 1976 "proof" of "The Four Color Map Theorem", Appel & Haken, U. of IL, (or Robertson, et al, GA Tech, followup)there was mention of P.G. Tait's 1880 "three-edge coloring" -- which apparently led them astray in setting their "rules". Wikipedia has considerable material about Tait (including "quad...", Calculus topic), but NOTHING on "t-e.c."). I wonder if Cambridge 3-vol. production discussed it at all. [See: MATHEMATICS: From the Birth of Numbers, by Prof. Jan Gullberg (ca 2001).] As of now, I can only speculate (indicated above). —Preceding unsigned comment added by Crochety77 (talk • contribs) 22:50, 8 August 2008 (UTC)

Do you mean Tait's conjecture? Jkasd 17:20, 9 August 2008 (UTC)

Could be, but it's also likely that he means Tait's 1880 proof that 4CT is equivalent to the nonexistence of a nonplanar snark. —David Eppstein (talk) 16:51, 11 August 2008 (UTC)