Thread for NonCA Academic Questions
 gameoflifemaniac
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Re: Thread for NonCA Academic Questions
What's the definition of trivial?
https://www.youtube.com/watch?v=q6EoRBvdVPQ
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 Apple Bottom
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Re: Thread for NonCA Academic Questions
A mathematician is presenting some new results to a colleague. His colleague points out that he did not prove a theorem; the mathematician replies that the proof is trivial. At his colleague's request, he starts proving the theorem anyway. After two hours, he's done; all the room's blackboards are filled with arcane formulae and obscure symbols. His colleague nods, strokes his beard thoughtfully, and agrees that yes, it IS trivial.gameoflifemaniac wrote:What's the definition of trivial?
If you speak, your speech must be better than your silence would have been. — Arabian proverb
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Re: Thread for NonCA Academic Questions
How does one reduce the infinite equation y = first root (x+ square root (x + cube root (x + fourth root (x + fifth root( x + nth root(x + ...
,where n is the number of radicals in the series up to and including that point?
,where n is the number of radicals in the series up to and including that point?
Code: Select all
x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!
 A for awesome
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Re: Thread for NonCA Academic Questions
Anyone have any idea how to find (EDIT for clarification: or at least approximate) realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r? I'm trying to define tetration for noninteger real values without reading exactly why it's impossible first. (Next I'd need to find g(r) so that g(g(r)) = f_x(r), h(r) so that h(h(r)) = g_x(r), etc..) I'm not sure whether it's possible without some other constraint, though  f(f(r)) = r has uncountably infinite solutions, and in any case I can't think of a nonredundant constraint for the other one apart from infiniteorder differentiability on some open interval, which still doesn't yield finitely many solutions.
Hopefully I said all that right.
Hopefully I said all that right.
Last edited by A for awesome on October 10th, 2017, 6:12 pm, edited 1 time in total.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
Re: Thread for NonCA Academic Questions
x^(r/2)?A for awesome wrote:Anyone have any idea how to find realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
 A for awesome
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Re: Thread for NonCA Academic Questions
Maybe I wasn't clear enough, but the way I'm constructing it, that would give f(f(r)) = x^((x^(r/2))/2) ≠ x^(r/2).Macbi wrote:x^(r/2)?A for awesome wrote:Anyone have any idea how to find realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
http://conwaylife.com/wiki/A_for_all
Aidan F. Pierce
 Apple Bottom
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Re: Thread for NonCA Academic Questions
For starters, see e.g. https://en.wikipedia.org/wiki/Halfexponential_function , which has some links/references that may be helpful.A for awesome wrote:Anyone have any idea how to find (EDIT for clarification: or at least approximate) realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
If you speak, your speech must be better than your silence would have been. — Arabian proverb
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Re: Thread for NonCA Academic Questions
What happen if oil (black gold) became worthless because of the invention of a magic infinite oil machine? (E.g. free oil for people who want it)
What would happen to "oil kingdoms" such as the Middle East?
What would happen to "oil kingdoms" such as the Middle East?
Airy Clave White It Nay
(Check gen 2)
Code: Select all
x = 17, y = 10, rule = B3/S23
b2ob2obo5b2o$11b4obo$2bob3o2bo2b3o$bo3b2o4b2o$o2bo2bob2o3b4o$bob2obo5b
o2b2o$2b2o4bobo2b3o$bo3b5ob2obobo$2bo5bob2o$4bob2o2bobobo!
Re: Thread for NonCA Academic Questions
x^sqrt(r) may work for positive r:Macbi wrote:x^(r/2)?A for awesome wrote:Anyone have any idea how to find realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
f(x)=x^sqrt(r)
f(f(x))=(x^sqrt(r))^sqrt(r)=x^(sqrt(r)*sqrt(r))=x^r
Still drifting.
Re: Thread for NonCA Academic Questions
Oh yeah, I was doubly wrong.Bullet51 wrote:x^sqrt(r) may work for positive r:Macbi wrote:x^(r/2)?A for awesome wrote:Anyone have any idea how to find realvalued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
f(x)=x^sqrt(r)
f(f(x))=(x^sqrt(r))^sqrt(r)=x^(sqrt(r)*sqrt(r))=x^r
Re: Thread for NonCA Academic Questions
Is it possible for star polygons, such as {3/2} and {5/3}, to exist on the hyperbolic plane?
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
Re: Thread for NonCA Academic Questions
Also, are there any other numbers where n, n+2, n+6, n+8, n+90, n+92, 9+96 and n+98 are all prime (or as I call them, twin primequadruplets)?
The only numbers I can find where this works are 11 and 101.
The only numbers I can find where this works are 11 and 101.
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
Re: Thread for NonCA Academic Questions
It's possible to find closer pairs of PQs, e.g. 1006301 + {0, 2, 6, 8, 30, 32, 36, 38} are all prime, so that should define a 'twin prime quadruplet'. The first few such twin prime quadruplets are shown at https://oeis.org/A059925 if you're interested.muzik wrote:Also, are there any other numbers where n, n+2, n+6, n+8, n+90, n+92, 9+96 and n+98 are all prime (or as I call them, twin primequadruplets)?
The only numbers I can find where this works are 11 and 101.
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Re: Thread for NonCA Academic Questions
So those can exist...
If these are to be the twin primequadruplets, then were we to extrapolate from the names of prime numbers differing by an even number n (twin for 2, cousin for 4, sexy for 6, etc) and take the distance between the first number in each prime quadruplet in the grouping to be 15n, would this result in the originally proposed type of grouping be called "sexy primequadruplets"? Or is this paragraph a load of nonsense?
If these are to be the twin primequadruplets, then were we to extrapolate from the names of prime numbers differing by an even number n (twin for 2, cousin for 4, sexy for 6, etc) and take the distance between the first number in each prime quadruplet in the grouping to be 15n, would this result in the originally proposed type of grouping be called "sexy primequadruplets"? Or is this paragraph a load of nonsense?
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
Re: Thread for NonCA Academic Questions
15641 fits; I can't find any other number that does but I'm sure that they're out there.muzik wrote:Also, are there any other numbers where n, n+2, n+6, n+8, n+90, n+92, 9+96 and n+98 are all prime (or as I call them, twin primequadruplets)?
The only numbers I can find where this works are 11 and 101.
Re: Thread for NonCA Academic Questions
Nothing came of the OEIS, but punching "11 101 15641" in Google including the quotation marks led to a German paper which seems to give multiple more numbers which look like they might fit this sequence.
11, 101, 15641, 3512981, 6655541, 20769311, 26919791, 41487071, 71541641, 160471601, 189425981
11, 101, 15641, 3512981, 6655541, 20769311, 26919791, 41487071, 71541641, 160471601, 189425981
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
Re: Thread for NonCA Academic Questions
I get the following exhaustive enumeration of values below 10^9:muzik wrote:Nothing came of the OEIS, but punching "11 101 15641" in Google including the quotation marks led to a German paper which seems to give multiple more numbers which look like they might fit this sequence.
11, 101, 15641, 3512981, 6655541, 20769311, 26919791, 41487071, 71541641, 160471601, 189425981
11, 101, 15641, 3512981, 6655541, 20769311, 26919791, 41487071, 71541641, 160471601, 189425981, 236531921, 338030591, 409952351, 423685721, 431343461, 518137091, 543062621, 588273221, 637272191, 639387311, 647851571, 705497951, 726391571, 843404201, 895161341, 958438751, 960813851, 964812461, 985123961
What do you do with ill crystallographers? Take them to the monoclinic!
Re: Thread for NonCA Academic Questions
I guess that's pretty neat.
The paper doesn't mention any 60 or 150 type gaps, though. Can they be proved to not exist, or are they just really big?
As for the 3/2 "star triangle" question, here's roughly what I'd assume such a figure to look like:
Are there any more well known names for this shape (aside from "fidget spinner" or "that one weird shape you always find drawn in old textbooks")?
5/3:
The paper doesn't mention any 60 or 150 type gaps, though. Can they be proved to not exist, or are they just really big?
As for the 3/2 "star triangle" question, here's roughly what I'd assume such a figure to look like:
Code: Select all
x = 80, y = 74, rule = B3ai4a/S3ai4a5ai6ac7c8
62b9ob2o$59b3o12bo$58bo16b2o$56b2o19bo$54b2o22bo$53bo24bo$52bo26bo$51b
o27bo$50bo28bo$49bo29bo$23b13o12bo30bo$19b4o12b3o9bo31bo$17b2o19bo7bo
32bo$16bo22bo6bo32bo$13b3o24bo4bo33bo$13bo27bo3bo33bo$11b2o28bo2bo34bo
$9b2o31b2o35bo$8bo33b2o35bo$8bo33b2o35bo$7bo33bo2bo33bo$6bo34bo3bo32bo
$5bo34bo5bo31bo$5bo34bo6bo29bo$4bo34bo7bo29bo$3bo35bo8bo27bo$2bo35bo
10bo25bo$bo36bo10bo24bo$bo35bo12bo23bo$o36bo13bo21bo$o36bo13bo20bo$o
35bo15bo18bo$o35bo15bo17bo$o34bo17bo15bo$o34bo17bo14bo$o33bo19bo11b2o$
o33bo19bo9b2o$bo31bo21bo5b3o$bo31bo22bob3o$b2o29bo21b4o$2bo29bo17b4o3b
o$3bo28bo14b3o7bo$4b3o25bo7b7o10bo$7bo23bob7o17bo$8b25o24bo$31bo25bo$
30bo27bo$30bo27bo$29bo28bo$29bo28bo$28bo29bo$28bo29bo$28bo29bo$28bo29b
o$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo28bo$28bo
28bo$28bo27bo$28bo26bo$28bo24b2o$28bo23bo$28bo22bo$29bo20bo$29bo19bo$
29bo17b2o$29bo16bo$30bo13b2o$30bo11b2o$31b11o!
5/3:
Code: Select all
x = 115, y = 123, rule = B3ai4a/S3ai4a5ai6ac7c8
39b11o$35b4o11b2o$32b3o16b2o22b12o$29b3o21bo11b11o10b5o$27b2o25b2o7b2o
25b3o$26bo28bo4b3o29b2o$23b3o30bo2bo33b3o$22bo33bobo36b2o$21bo34b2o38b
2o$19b2o33b5o38b2o$17b3o33b2o2bobo38b2o$16bo36bo3b2obo38bo$15bo36bo7bo
38bo$14bo36bo9bo37bo$13bo37bo9bo37bo$12bo37bo10bo37bo$11bo37bo12bo36bo
$10b2o36bo13bo36bo$10bo37bo14bo35bo$9bo38bo14bo35bo$9bo38bo14bo35bo$9b
o37bo15bo35bo$9b2o36bo16bo34bo$10bo35bo17bo34bo$10bo35bo17bo34bo$10bo
35bo17bo34bo$10bo34bo18bo33b2o$10bo34bo18bo33bo$10b2o15b45o26bo$11bo
12b3o18bo19bo6b10o15bo$11bo10b2o21bo19bo16b3o11bo$12bo7b2o22bo20bo19bo
3bo5b2o$13bo5b2o23bo20bo20b2o2b3o2bo$14b2o2b2o24bo20bo21bo5b3o$16b2o
26bo21bo20bo5b5o$17b2o24bo22bo20b2o4bo3b3o$16bobo24bo22bo21bo3b2o5b2o$
15bo3bo23bo23bo20bo2bo8b2o$14b2o4bo22bo23bo19bo2b2o10bo$14bo6bo20bo24b
o19bo2bo12bo$13b2o7bo19bo24bo18bo2bo13b2o$13bo9b2o16bo26bo16bo2b2o14bo
$12bo12bo15bo26bo15b2o2bo16bo$11b2o13bo14bo26bo14bo3b2o17b2o$10b2o15bo
13bo27bo11b2o3b2o19b2o$10bo17bo11bo28bo10bo4bo22b2o$9bo19bo10bo28bo9bo
4bo24b2o$8b2o20bo9bo29bo8bo30bo$8bo22bo8bo29bo6b2o31b2o$7bo24bo7bo29bo
5bo34bo$6bo26b2o4bo31bo3bo35bo$6bo28bo3bo31bob2o36bo$5bo30bobo32bobo
37b2o$4bo31bobo33bo39bo$4bo32b2o32b2o39bo$3bo33bobo31b2o39bo$3bo33bo2b
o29bobo39bo$2bo34bo3bo27bo3bo39bo$2bo34bo4bo25bo4bo39bo$2bo34bo5bo23bo
5bo39bo$bo35bo6bo22bo5bo40bo$bo34bo8bo20bo6bo40bo$bo34bo9bo19bo7bo39bo
$bo34bo10bo17bo8bo39bo$bo34bo11bo16bo9bo38bo$o34bo13bo14bo10bo38bo$o
34bo14bo12bo11bo38bo$o34bo14bo12bo11bo38bo$o34bo15b2o10bo11bo38bo$o33b
o18bo8bo12bo38bo$o33bo19bo6bo13bo38bo$o33bo20bo4bo14bo38bo$o33bo21bo3b
o14bo38bo$o33bo22b3o15bo38bo$o33bo23b2o15bo38bo$o33bo22bobo15bo37bo$o
33bo21bo3b2o13bo36b2o$o33bo20bo6bo12bo36bo$o33b2o18bo8b2o10bo35bo$o33b
2o17b2o10b3o7bo35bo$o34bo15b2o14b9o34bo$o35bo12b2o17b2o6bo32bo$o35b2o
9b2o21b2o4bo30b2o$o36b2o5b3o24b3o3bo27b2o$o35bo2b5o30b5o24b2o$o35bo4bo
35b5o14b7o$bo34bo2b2o37bo3b15o$bo35b2o39bo$bo31b5o40bo$2bo28b2o4bo41bo
$2bo26b3o5b2o40bo$2bo22b4o9bo41bo$3bo17b5o12bo41bo$3b2o16bo16bo42bo$4b
2o14bo18bo41bo$6b2o9b3o19bo41b2o$8b10o21bo42bo$39bo42bo$39bo42bo$39bo
42bo$39bo42bo$39b2o41bo$40bo41bo$40bo41bo$40bo41bo$40bo41bo$41bo40bo$
41bo40bo$42bo39bo$42bo39bo$43bo37b2o$43bo37bo$44bo36bo$44bo35bo$45bo
34bo$45bo34bo$46bo32b2o$46bo32bo$47bo30b2o$47b2o28b2o$48b2o24b4o$50b2o
15b8o$52b15o!
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
 toroidalet
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Re: Thread for NonCA Academic Questions
A trefoil, maybe?muzik wrote:Are there any more well known names for this shape (aside from "fidget spinner" or "that one weird shape you always find drawn in old textbooks")?
Could be likened to a biohazard or radiation symbol but these aren't topologically identical.
Also here's an interesting conjecture with no practical value: If you set f(x) as d/dx ln x, does the function g(x)=f(x)ln(ax) has a local maximum at x=1/a?
"Build a man a fire and he'll be warm for a day. Set a man on fire and he'll be warm for the rest of his life."
Terry Pratchett
Terry Pratchett
Re: Thread for NonCA Academic Questions
Seems as though "trefoil knot" fits this shape well enough. Can't seem to find any shapes that match the 5/3 shape though.
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!

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Re: Thread for NonCA Academic Questions
The closest think I could think of was a Spirograph.
Code: Select all
x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!
Re: Thread for NonCA Academic Questions
Is 11 truly the only number where n, n+2, n+6, n+8, n+90, n+92, 9+96, n+98, n+180, n+182, n+186 and n+188 are all prime? If there are any more such numbers out there I will be surprised.
Bored of using the Moore neighbourhood for everything? Introducing the Range2 von Neumann isotropic nontotalistic rulespace!
Re: Thread for NonCA Academic Questions
Conjecturally, and heuristically, there are infinitely many.muzik wrote:Is 11 truly the only number where n, n+2, n+6, n+8, n+90, n+92, 9+96, n+98, n+180, n+182, n+186 and n+188 are all prime? If there are any more such numbers out there I will be surprised.
What do you do with ill crystallographers? Take them to the monoclinic!

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Re: Thread for NonCA Academic Questions
Probably not, as calcyman said, but none that I could find up to 10^8.muzik wrote:Is 11 truly the only number where n, n+2, n+6, n+8, n+90, n+92, 9+96, n+98, n+180, n+182, n+186 and n+188 are all prime? If there are any more such numbers out there I will be surprised.
I and wildmyron manage the 5S project, which collects all known spaceship speeds in Isotropic Nontotalistic rules.
Things to work on:
 Find a (7,1)c/8 ship in a Nontotalistic rule
 Finish a rule with ships with period >= f_e_0(n) (in progress)
Things to work on:
 Find a (7,1)c/8 ship in a Nontotalistic rule
 Finish a rule with ships with period >= f_e_0(n) (in progress)
 gameoflifemaniac
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 Location: There too
Re: Thread for NonCA Academic Questions
How are all the bits converted in to music?
https://www.youtube.com/watch?v=q6EoRBvdVPQ
One big dirty Oro. Yeeeeeeeeee...
One big dirty Oro. Yeeeeeeeeee...