Wikipedia:Reference desk/Archives/Mathematics/2012 September 14

= September 14 =

Suspected fault in article on wikipedia
Hey

I'm not certain if this is the correct place for this, but then again someone will see it and perhaps edit it.

http://en.wikipedia.org/wiki/Ancient_Roman_units_of_measurement#Liquid_measures

In the above link it is stated that a cubic foot is equal to 8 *unitX*, while a half-foot cubed is equal to 4 *unitX*. I do believe a half-foot cubed should be a fourth of a foot cubed, not a half as stated here.

Pleasant day to you, Sirs and Madames — Preceding unsigned comment added by 94.22.125.155 (talk) 08:19, 14 September 2012 (UTC)


 * My reading of the section that you link to is that it says a "amphora quadrantal" has a volume of one cubic foot, a "congius" is a half-foot cubed (i.e. its volume is equal to a cube that is half a foot in each dimension), and in the table it says that an amphora quadrantal is equal to 8 congii. This seems correct to me. A half-foot squared would be a quarter of a square foot, but a half-foot cubed is one eighth of a cubic foot because (1/2) x (1/2) x (1/2) = 1/8. Gandalf61 (talk) 08:49, 14 September 2012 (UTC)

Growth Dirichlet Kernel
First of all, if you can show $$\lim_{n \rightarrow \infty}\frac{f(n)}{g(n)}=1$$ where $$f$$ and $$g$$ are positive and $$\forall n \in \mathbb{N} f(n) < g(n)$$ is it true that there then exists a constant $$c >0 $$ such that $$\forall n \in \mathbb{N} f(n) > cg(n)$$?

If so, show that $$\lim_{n \rightarrow \infty} \frac{\int_{-\pi}^\pi \min\left(n^2\left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|,1\right)\left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|dy}{\int_{-\pi}^\pi\left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|dy} = 1$$. I think equivalently $$\lim_{n \rightarrow \infty}\int_{-\pi}^\pi \left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|dy-\int_{-\pi}^\pi \min\left(n^2\left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|,1\right)\left|\frac{\sin((n+1/2)y)}{\sin(y/2)}\right|dy = 0$$. Widener (talk) 15:06, 14 September 2012 (UTC) For proof of the first part, I think you can take the $$n' \in \mathbb{N}$$ which minimizes the ratio between $$f$$ and $$g$$ (minimum exists because $$\lim_{n \rightarrow \infty}\frac{f(n)}{g(n)}=1$$) and then choose $$c < \frac{f(n')}{g(n')}$$ Widener (talk) 15:06, 14 September 2012 (UTC)

Regular structures in the prime spiral о_0
I tried to scale image of the prime spiral ( image). The result:

To repeat,just download aforenamed image, open it and try to reduce the size of the window of your image viewer (slowly).

Could you please explain me, why there (semi-)regular structures? Is it a property of the prime spiral or some software bug or some defect in interpolation/scale algorithm or what?

--Ewigekrieg (talk) 20:26, 14 September 2012 (UTC)


 * It's an aliasing effect, resulting from a combination of structure in the prime spiral with a crude method of image scaling. The structure in the prime spiral, by the way, is the topic of the so-called abc conjecture, although our article doesn't discuss it from that perspective.  (Note: I have taken the liberty of tweaking the layout of the question -- it was taking up way too much space.) Looie496 (talk) 20:39, 14 September 2012 (UTC)
 * Could you please explain the link between this structures and the conjecture? I'm not a professional mathematician. Also, why "islands" on the next to last section of image? --Ewigekrieg (talk) 20:53, 14 September 2012 (UTC)
 * Sorry, there is a story currently making the news that a Japanese mathematician, Shinichi Mochizuki, claims to have solved the abc conjecture, and some of the stories mention that it relates to the prime spiral, but that's the sum total of what I know about the prime spiral. See for example http://www.huffingtonpost.com/2012/09/12/abc-conjecture-shinichi-mochizuki-prime-numbers_n_1877692.html. Looie496 (talk) 21:54, 14 September 2012 (UTC)
 * That link doesn't actually claim a connection between the abc conjecture and the prime spiral and I haven't seen others do it. The prime spiral just seems like a nice image often used when otherwise non-photogenic prime numbers are discussed. PrimeHunter (talk) 22:21, 14 September 2012 (UTC)
 * At a first glance, the structures you are observing would require characterization of the aliasing properties of the scaling algorithm before proceeding any further. For example, whether the scaling algorithm is applied iteratively (pixels generated by a previous iteration being used, or the algorithm being applied to the raw data each time), what sampling/anti-aliasing algorithm is used (use of nearest pixel, or some blend of pixels in a region), variation of pixel interpolation at various interpolation positions (the weights assigned to each pixel in a region according to distance from nearest pixel: this can lead to Moiré patterns), and possible rounding errors/nonlinearities (e.g. choice of pixel may alternate between rows along a samping line between them, and when calculation of pixel intensity using weights). Only then can one answer whether there might be a subtle effect from the prime spiral. A few things are certain: there are aliasing effects occurring, and a "perfect" (Nyquist) interpolating filter will not produce these patterns. On the other hand, if the patterns are a genuine result of true aliasing (i.e not an artifact of the scaling implementation), these patterns would be of mathematical interest (as is the effect of the faint 45° patterns in the spiral). A quick test would be to see whether the same scaling produces similar behaviour from simple patterns, including random pixels (Tv snow). — Quondum 05:55, 15 September 2012 (UTC)