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April 2

Is a stretched circular arc similar to a circular arc of a different circle?

Consider a circular segment, such as the one bounded by circular arc S and chord C in the diagram below. Assume that θ is "small". If we stretch (scale) the circular segment horizontally, will the "stretched" circular arc still be (approximately?) similar to a circular arc on a circle with a different R? If that's the case, will the (approximate?) similarity get worse as the scaling factor increases? Is there a simple formula that can characterize how similar the stretched circular arc is to a true circular arc?

134.242.92.97 (talk) 03:26, 2 April 2024 (UTC)[reply]

By 'stretching', do you mean a non-uniform scaling, such that the size is enlarged along the chord while preserved along sagitta...? --CiaPan (talk) 14:48, 2 April 2024 (UTC)[reply]

Yes (scaling along the direction of the chord but not along the direction of the sagitta). --134.242.92.97 (talk) 16:25, 2 April 2024 (UTC)[reply]

When you scale a circle, the resulting figure is an ellipse. An elliptical segment is not similar to a circular one in geometric sense. Of course the elliptic segment can be approximated with a circular one, so in a common speech they can be called 'similar', but I suppose that's not an appropriate use of the word here, at math ref.desk. --CiaPan (talk) 17:36, 2 April 2024 (UTC)[reply]

Note that the question assumes θ is "small" and the "similarity" may be approximate. Is there a simple answer to the question in terms of some intuitive similarity metric? (A similarity metric can be defined, in different ways, in terms of the deviation of the stretched arc relative to the best-fitting circular arc.) --134.242.92.97 (talk) 19:06, 2 April 2024 (UTC)[reply]

The area between the two curves (the stretched circle and the circle approximating it) could be used as a measure of the error. If the stretching factor is

f_{s}

the area of the stretched segment is

{\begin{aligned}A_{s}&={\frac {1}{2}}f_{s}(\theta R^{2}-dc)\\&=f_{s}\left(\arcsin \left({\frac {c}{2R}}\right)R^{2}-{\frac {(R-h)c}{2}}\right)\\&={\frac {f_{s}}{4}}\left(\arcsin \left({\frac {4h}{4h^{2}+c^{2}}}c\right)\left({\frac {4h^{2}+c^{2}}{4h}}\right)^{2}+{\frac {4h^{2}-c^{2}}{4h}}c\right)\end{aligned}}

If the approximating circle is chosen to go through the end-points and the mid-point off the stretched arc, then its radius

R_{a}

satisfies

R_{a}^{2}=(R_{a}-h)^{2}+\left({\frac {f_{s}c}{2}}\right)^{2}

so

R_{a}={\frac {4h^{2}+f_{s}^{2}c^{2}}{8h}}

and the area of the approximating segment is

{\begin{aligned}A_{a}&=\arcsin \left({\frac {f_{s}c}{2R_{a}}}\right)R_{a}^{2}-{\frac {(R_{a}-h)f_{s}c}{2}}\\&={\frac {1}{4}}\left(\arcsin \left({\frac {4h}{4h^{2}+f_{s}^{2}c^{2}}}f_{s}c\right)\left({\frac {4h^{2}+f_{s}^{2}c^{2}}{4h}}\right)^{2}+{\frac {4h^{2}-f_{s}^{2}c^{2}}{4h}}f_{s}c\right)\end{aligned}}

and the area between the curves is

A_{a}-A_{s}={\frac {1}{4}}\left(\arcsin \left({\frac {4h}{4h^{2}+f_{s}^{2}c^{2}}}f_{s}c\right)\left({\frac {4h^{2}+f_{s}^{2}c^{2}}{4h}}\right)^{2}-f_{s}\arcsin \left({\frac {4h}{4h^{2}+c^{2}}}c\right)\left({\frac {4h^{2}+c^{2}}{4h}}\right)^{2}-{\frac {c^{3}f_{s}(f_{s}^{2}-1)}{4h}}\right)

or given that the angle is small, approximating

\arcsin(x)\approx x+{\frac {x^{3}}{6}}

A_{a}-A_{s}\approx {\frac {2c^{3}f_{s}h^{3}(f_{s}^{2}-1)}{3(4h^{2}+f_{s}^{2}c^{2})(4h^{2}+c^{2})}}={\frac {f_{s}c^{3}h(f_{s}^{2}-1)}{96RR_{a}}}

(barring mistakes) catslash (talk) 00:01, 3 April 2024 (UTC)[reply]

Thanks for the reply. It seems that your formula is for the difference in area between the two segments, but not the area between the two curves. --134.242.92.97 (talk) 17:12, 3 April 2024 (UTC)[reply]

That is true, but the the two segments have the same lower boundary, namely the stretched arc, and so the difference in their areas is area between their upper boundaries, i.e. the area between the two curves. But in any case, Lambian's answer is better. catslash (talk) 22:30, 4 April 2024 (UTC)[reply]

A suitable measure for the dissimilarity of two curves is the Fréchet distance. --Lambiam 21:04, 3 April 2024 (UTC)[reply]

Then if the approximating circle is again chosen to go through the end-points and the mid-point off the stretched arc

F=f_{s}{\sqrt {R^{2}+{\frac {(f_{s}^{2}-1)(2R-h)^{2}}{4}}}}+{\frac {h(f_{s}^{2}-1)-2Rf_{s}^{2}}{2}}

(this gives

\left.F\right|_{f_{s}=1}=0

and

\left.F\right|_{f_{s}=0}=-{\frac {h}{2}}

as expected, so could be correct). However, it is possible to get a smaller

F

by choosing an approximation which passes above/below (

f_{s}\gtrless 1

)the mid-point of the stretched arc rather than through it. catslash (talk) 22:22, 4 April 2024 (UTC)[reply]

For which n, A269252(n) is -1?

[1] proves that A269254(110) and A269252(34) are both -1, and there is a sequence for the n such that A269254(n) = -1: A333859, but there seems to be no OEIS sequence for the n such that A269252(n) = -1, and A269254 and A269252 are similar sequence, so for which n, A269252(n) is -1? 118.170.19.90 (talk) 09:41, 2 April 2024 (UTC)[reply]

A269252(n) = -1 for n = 1, 2, 34, 53., but there may not be a sequence 1, 2, 34, 53, ... Bubba73 ^{You talkin' to me?} 20:04, 2 April 2024 (UTC)[reply]

Could you compute the terms <= 1000? Thanks. A333859 has the terms <= 1500. 49.217.123.95 (talk) 08:21, 9 April 2024 (UTC)[reply]

April 4

K-triviality using conditional complexity?

To account for length, the definition of K-trivial set makes the complexity of the length part of the upper-bound on the complexity of the string. Has anyone determined what happens if one instead asks for a bound on the length-conditional complexity? This could be done with either plain or prefix-free Kolmogorov complexity. JumpDiscont (talk) 00:12, 4 April 2024 (UTC)[reply]

If there's a constant bound on the length-conditioned complexity, then the set is computable. For a bound that tends to infinity (maybe

K(n)

?), this is similar to c.e.-traceability, so I suspect it won't line up exactly with

K

-triviality.

For research level math like this, you'll probably have better luck on mathoverflow.--Antendren (talk) 10:04, 6 April 2024 (UTC)[reply]

April 8

Constructing Wieferich numbers according to Agoh, Dilcher and Skula

I am trying to understand how to use the method in https://doi.org/10.1006/jnth.1997.2162 to construct composite base-a Wieferich numbers from a given integer a > 1 and a base-a Wieferich prime p. In section 6. EXAMPLES the authors use Theorem 5.5 to construct the list of base-2 Wieferich numbers based on the two known Wieferich primes 1093 and 3511. They compute the prime factorizations of 1092 and 3510 and apply Theorem 5.5 to obtain the 104 known base-2 Wieferich numbers. How does this work for arbitrary Wieferich primes p and bases a? For example, suppose I have the prime number p = 5209 which is a Wieferich prime to base a = 1359624 (see https://oeis.org/A143548 comment no. 3 by T. D. Noe). 5208 = 2³ * 3 * 7 * 31. How does one obtain the base-1359624 Wieferich numbers generated by 5209 with Theorem 5.5? I also find the notation in the paper a bit confusing: q(a, m) is an Euler quotient per definition 1.2, but I don't really understand the definition of σ(a, p) at the top of p. 46. Is this simply the multiplicative order of a (mod p)? Toshio Yamaguchi (talk) 11:40, 8 April 2024 (UTC)[reply]

Is your question related to [2]? Which is the largest known Wieferich numbers to bases b<=25 2402:7500:916:7991:1CB3:D7F2:BD28:3F3A (talk) 08:11, 9 April 2024 (UTC)[reply]

Mhm, I guess it is related. If I understand correctly, Theorem 2 in that paper could be used to generate a Wieferich number w based on a given Wieferich prime p, although I have yet to understand how exactly that would work. Toshio Yamaguchi (talk) 09:22, 9 April 2024 (UTC)[reply]

A further question: On p. 2 of the Akbari, Siavashi paper there is a following statement I do not understand: "For a prime p and positive integer n, we denote the largest power of p in n by v_p(n)". What does "power of p in n" mean? Toshio Yamaguchi (talk) 09:38, 9 April 2024 (UTC)[reply]

April 9

What is the smallest positive integer which is known to not divide any (even or odd) perfect number?

What is the smallest positive integer which is known to not divide any (even or odd) perfect number? Also, how many positive integers <= 400 are known to not divide any (even or odd) perfect number? 49.217.123.95 (talk) 08:05, 9 April 2024 (UTC)[reply]

Is it possible to use all n-cubes with n <= 5 assembled into a 2331 rectangular cuboid?

Is it possible to use all n-cubes with n <= 5 (reflections counted as distinct), with totally 186 cubes, assembled into a 2*3*31 rectangular cuboid?

Also, for which positive integer N, it is possible to use all n-cubes with n <= N (reflections counted as distinct), assembled into a rectangular cuboid? 2402:7500:916:7991:1CB3:D7F2:BD28:3F3A (talk) 08:49, 9 April 2024 (UTC)[reply]

There are 11-polycubes with holes in the center, and it's impossible to fill this hole with anything but a single cube. There are several such shapes, and you only get one single cube, so N≥11 is impossible. You run into similar problems with holes when you try to tile with heptominos. You could stipulate that the pieces with holes not be included, but that would be a different problem. For N=5, it seems to me that best (or at least the most fun) approach would be to whip up a set with a 3-D printer and experiment. Sometimes a parity argument can give a simple impossibility proof, but that seems unlikely in this case. --RDBury (talk) 12:56, 9 April 2024 (UTC)[reply]