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gameoflifemaniac
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What's the definition of trivial?
One big dirty Oro. Yeeeeeeeeee...

Apple Bottom
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Contact:

gameoflifemaniac wrote:What's the definition of trivial?
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.
If you speak, your speech must be better than your silence would have been. — Arabian proverb

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Gamedziner
Posts: 796
Joined: May 30th, 2016, 8:47 pm
Location: Milky Way Galaxy: Planet Earth

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?

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 Posts: 1901 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: Thread for Non-CA Academic Questions Anyone have any idea how to find (EDIT for clarification: or at least approximate) real-valued 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 non-integer 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 non-redundant constraint for the other one apart from infinite-order differentiability on some open interval, which still doesn't yield finitely many solutions. 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 Macbi Posts: 699 Joined: March 29th, 2009, 4:58 am ### Re: Thread for Non-CA Academic Questions A for awesome wrote:Anyone have any idea how to find real-valued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r? x^(r/2)? A for awesome Posts: 1901 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: Thread for Non-CA Academic Questions Macbi wrote: A for awesome wrote:Anyone have any idea how to find real-valued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r? x^(r/2)? 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). 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 Apple Bottom Posts: 1027 Joined: July 27th, 2015, 2:06 pm Contact: ### Re: Thread for Non-CA Academic Questions A for awesome wrote:Anyone have any idea how to find (EDIT for clarification: or at least approximate) real-valued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r? For starters, see e.g. https://en.wikipedia.org/wiki/Half-exponential_function , which has some links/references that may be helpful. If you speak, your speech must be better than your silence would have been. — Arabian proverb Catagolue: Apple Bottom • Life Wiki: Apple Bottom • Twitter: @_AppleBottom_ Proud member of the Pattern Raiders! Saka Posts: 3138 Joined: June 19th, 2015, 8:50 pm Location: In the kingdom of Sultan Hamengkubuwono X ### Re: Thread for Non-CA 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? Airy Clave White It Nay Code: Select all x = 17, y = 10, rule = B3/S23 b2ob2obo5b2o$11b4obo$2bob3o2bo2b3o$bo3b2o4b2o$o2bo2bob2o3b4o$bob2obo5b
o2b2o$2b2o4bobo2b3o$bo3b5ob2obobo$2bo5bob2o$4bob2o2bobobo!

(Check gen 2)

Bullet51
Posts: 536
Joined: July 21st, 2014, 4:35 am

Macbi wrote:
A for awesome wrote:Anyone have any idea how to find real-valued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
x^(r/2)?
x^sqrt(r) may work for positive r:
f(x)=x^sqrt(r)
f(f(x))=(x^sqrt(r))^sqrt(r)=x^(sqrt(r)*sqrt(r))=x^r
Still drifting.

Macbi
Posts: 699
Joined: March 29th, 2009, 4:58 am

Bullet51 wrote:
Macbi wrote:
A for awesome wrote:Anyone have any idea how to find real-valued solutions f(r) to the equation f(f(r)) = x^r for given real x and a real variable r?
x^(r/2)?
x^sqrt(r) may work for positive r:
f(x)=x^sqrt(r)
f(f(x))=(x^sqrt(r))^sqrt(r)=x^(sqrt(r)*sqrt(r))=x^r
Oh yeah, I was doubly wrong.

muzik
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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 Range-2 von Neumann isotropic non-totalistic rulespace!

muzik
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Location: Scotland

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 prime-quadruplets)?

The only numbers I can find where this works are 11 and 101.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!

calcyman
Posts: 2096
Joined: June 1st, 2009, 4:32 pm

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 prime-quadruplets)?

The only numbers I can find where this works are 11 and 101.
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.
What do you do with ill crystallographers? Take them to the mono-clinic!

muzik
Posts: 3499
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

So those can exist...

If these are to be the twin prime-quadruplets, 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 prime-quadruplets"? Or is this paragraph a load of nonsense?
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!

Rocknlol
Posts: 83
Joined: April 15th, 2012, 9:06 am

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 prime-quadruplets)?

The only numbers I can find where this works are 11 and 101.
15641 fits; I can't find any other number that does but I'm sure that they're out there.

muzik
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Location: Scotland

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
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!

calcyman
Posts: 2096
Joined: June 1st, 2009, 4:32 pm

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
I get the following exhaustive enumeration of values below 10^9:

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 mono-clinic!

muzik
Posts: 3499
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

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:

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!  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: 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 Range-2 von Neumann isotropic non-totalistic rulespace! toroidalet Posts: 1019 Joined: August 7th, 2016, 1:48 pm Location: my computer Contact: ### Re: Thread for Non-CA Academic Questions 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")? A trefoil, maybe? 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 muzik Posts: 3499 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### Re: Thread for Non-CA 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 Range-2 von Neumann isotropic non-totalistic rulespace! Gamedziner Posts: 796 Joined: May 30th, 2016, 8:47 pm Location: Milky Way Galaxy: Planet Earth ### Re: Thread for Non-CA 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!


muzik
Posts: 3499
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

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 Range-2 von Neumann isotropic non-totalistic rulespace!

calcyman
Posts: 2096
Joined: June 1st, 2009, 4:32 pm

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.
Conjecturally, and heuristically, there are infinitely many.
What do you do with ill crystallographers? Take them to the mono-clinic!

AforAmpere
Posts: 1049
Joined: July 1st, 2016, 3:58 pm

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.
Probably not, as calcyman said, but none that I could find up to 10^8.
I and wildmyron manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules.

Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule
- Finish a rule with ships with period >= f_e_0(n) (in progress)

gameoflifemaniac
Posts: 774
Joined: January 22nd, 2017, 11:17 am
Location: There too