Testwiki:Reference desk/Archives/Mathematics/2023 January 27

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January 27

Diameter of the circumscribed circle of a 11-sided Reuleaux polygon

For a Reuleaux triangle of width 1, the diameter of its circumscribed circle is approximately 1.15. (I don't have an analytical solution for this. I just drew it out and measured it.)

For an 11-sided Reuleaux polygon of width 1, what would the diameter of its circumscribed circle be?

(This is related to a question I asked on the Humanities reference desk[1]. Basically I'm trying approximate the size of the circumscribed circle of a Canadian Loonie, which happens to be an 11-sided Reuleaux polygon.) Helian James (talk) 04:56, 27 January 2023 (UTC)

What you want is the ratio of the diameter of a circle circumscribed about a regular 11-gon to its long diagonal. Or if you like, the ratio of the two longest diagonals in a regular 22-gon. This should be $2/|e^{\pi i/11}+1|=\sec \frac{\pi }{22}\approx 1.0103.$ jacobolus (t) 05:21, 27 January 2023 (UTC)

Thank you!!!Helian James (talk) 05:26, 27 January 2023 (UTC)

So if the width of the loonie is indeed 26.5 mm, then the circumcircle would have diameter a bit under 26.8 mm. –jacobolus (t) 05:28, 27 January 2023 (UTC)

P-recursive equation definition

I don't really understand the definition of P-recursive equation. It states that it is a linear equation but the coefficients are polynomials (or specially, sequences that are representable by polynomials)? How would that work?  AltoStev (talk) 17:27, 27 January 2023 (UTC)

A standard linear recurrence relation would look like $a{y}_n+b{y}_{n+1}+c{y}_{n+2}=d$, where $a,b,c,d$ are constants. In a p-recursive recurrence relation, $a,b,c,d$ are instead polynomials in $n$. That's it.--2600:4040:7B33:6E00:5DB1:E81C:2118:36DD (talk) 18:18, 27 January 2023 (UTC)

Do you have any idea why the article does not use subscripts for the $y$ sequence? And are there uses in which $f$ is not a polynomial? The presentation

$\sum _{k=0}^r{P}_k(n)y_{n+k}=Q(n)$

seems easier to grasp. One or two examples to illustrate the definition wouldn't be misplaced.  --Lambiam 00:04, 28 January 2023 (UTC)

I'd never seen P-recursive equations before clicking on the link in the OP, so I can't answer these questions.--2600:4040:7B33:6E00:B123:9399:E930:26DA (talk) 00:30, 28 January 2023 (UTC)

After looking at definitions in the literature, instead of defining "P-recursive equations", most define the notion of a "P-recursive sequence" as one obeying a homogeneous P-recursive relation, using subscripts for the sequence elements (e.g. here, where it is also stated that the nonhomogeneous case can be transformed into a homogeneous one, which I think assumes a polynomial rhs). This source, also using subscript notation, first defines the notion of a "P-recursive" recurrence relation, allowing a non-zero rhs of the equation, but requires it to be a polynomial.  --Lambiam 09:29, 28 January 2023 (UTC)