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

gameoflifemaniac

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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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Apple Bottom

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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?
x = 81, y = 96, rule = LifeHistory58.2A$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! Gamedziner Posts: 698 Joined: May 30th, 2016, 8:47 pm Location: Milky Way Galaxy: Planet Earth ### 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 A for awesome Posts: 1803 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 ### 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)? Macbi Posts: 659 Joined: March 29th, 2009, 4:58 am ### 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 A for awesome Posts: 1803 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 ### 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! Apple Bottom Posts: 1023 Joined: July 27th, 2015, 2:06 pm ### 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? If you're the person that uploaded to Sakagolue illegally, please PM me. x = 17, y = 10, rule = B3/S23b2ob2obo5b2o$11b4obo$2bob3o2bo2b3o$bo3b2o4b2o$o2bo2bob2o3b4o$bob2obo5bo2b2o$2b2o4bobo2b3o$bo3b5ob2obobo$2bo5bob2o$4bob2o2bobobo!

(Check gen 2)

Saka

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Location: In the kingdom of Sultan Hamengkubuwono X

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.
Bullet51

Posts: 521
Joined: July 21st, 2014, 4:35 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.

Macbi

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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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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!
muzik

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

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!

calcyman

Posts: 2019
Joined: June 1st, 2009, 4:32 pm

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!
muzik

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Location: Scotland

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.
Rocknlol

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Joined: April 15th, 2012, 9:06 am

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!
muzik

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

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!

calcyman

Posts: 2019
Joined: June 1st, 2009, 4:32 pm

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:
x = 80, y = 74, rule = B3ai4a/S3ai4a5ai6ac7c862b9ob2o$59b3o12bo$58bo16b2o$56b2o19bo$54b2o22bo$53bo24bo$52bo26bo$51bo27bo$50bo28bo$49bo29bo$23b13o12bo30bo$19b4o12b3o9bo31bo$17b2o19bo7bo32bo$16bo22bo6bo32bo$13b3o24bo4bo33bo$13bo27bo3bo33bo$11b2o28bo2bo34bo$9b2o31b2o35bo$8bo33b2o35bo$8bo33b2o35bo$7bo33bo2bo33bo$6bo34bo3bo32bo$5bo34bo5bo31bo$5bo34bo6bo29bo$4bo34bo7bo29bo$3bo35bo8bo27bo$2bo35bo10bo25bo$bo36bo10bo24bo$bo35bo12bo23bo$o36bo13bo21bo$o36bo13bo20bo$o35bo15bo18bo$o35bo15bo17bo$o34bo17bo15bo$o34bo17bo14bo$o33bo19bo11b2o$o33bo19bo9b2o$bo31bo21bo5b3o$bo31bo22bob3o$b2o29bo21b4o$2bo29bo17b4o3bo$3bo28bo14b3o7bo$4b3o25bo7b7o10bo$7bo23bob7o17bo$8b25o24bo$31bo25bo$30bo27bo$30bo27bo$29bo28bo$29bo28bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo29bo$28bo28bo$28bo28bo$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: x = 115, y = 123, rule = B3ai4a/S3ai4a5ai6ac7c839b11o$35b4o11b2o$32b3o16b2o22b12o$29b3o21bo11b11o10b5o$27b2o25b2o7b2o25b3o$26bo28bo4b3o29b2o$23b3o30bo2bo33b3o$22bo33bobo36b2o$21bo34b2o38b2o$19b2o33b5o38b2o$17b3o33b2o2bobo38b2o$16bo36bo3b2obo38bo$15bo36bo7bo38bo$14bo36bo9bo37bo$13bo37bo9bo37bo$12bo37bo10bo37bo$11bo37bo12bo36bo$10b2o36bo13bo36bo$10bo37bo14bo35bo$9bo38bo14bo35bo$9bo38bo14bo35bo$9bo37bo15bo35bo$9b2o36bo16bo34bo$10bo35bo17bo34bo$10bo35bo17bo34bo$10bo35bo17bo34bo$10bo34bo18bo33b2o$10bo34bo18bo33bo$10b2o15b45o26bo$11bo12b3o18bo19bo6b10o15bo$11bo10b2o21bo19bo16b3o11bo$12bo7b2o22bo20bo19bo3bo5b2o$13bo5b2o23bo20bo20b2o2b3o2bo$14b2o2b2o24bo20bo21bo5b3o$16b2o26bo21bo20bo5b5o$17b2o24bo22bo20b2o4bo3b3o$16bobo24bo22bo21bo3b2o5b2o$15bo3bo23bo23bo20bo2bo8b2o$14b2o4bo22bo23bo19bo2b2o10bo$14bo6bo20bo24bo19bo2bo12bo$13b2o7bo19bo24bo18bo2bo13b2o$13bo9b2o16bo26bo16bo2b2o14bo$12bo12bo15bo26bo15b2o2bo16bo$11b2o13bo14bo26bo14bo3b2o17b2o$10b2o15bo13bo27bo11b2o3b2o19b2o$10bo17bo11bo28bo10bo4bo22b2o$9bo19bo10bo28bo9bo4bo24b2o$8b2o20bo9bo29bo8bo30bo$8bo22bo8bo29bo6b2o31b2o$7bo24bo7bo29bo5bo34bo$6bo26b2o4bo31bo3bo35bo$6bo28bo3bo31bob2o36bo$5bo30bobo32bobo37b2o$4bo31bobo33bo39bo$4bo32b2o32b2o39bo$3bo33bobo31b2o39bo$3bo33bo2bo29bobo39bo$2bo34bo3bo27bo3bo39bo$2bo34bo4bo25bo4bo39bo$2bo34bo5bo23bo5bo39bo$bo35bo6bo22bo5bo40bo$bo34bo8bo20bo6bo40bo$bo34bo9bo19bo7bo39bo$bo34bo10bo17bo8bo39bo$bo34bo11bo16bo9bo38bo$o34bo13bo14bo10bo38bo$o34bo14bo12bo11bo38bo$o34bo14bo12bo11bo38bo$o34bo15b2o10bo11bo38bo$o33bo18bo8bo12bo38bo$o33bo19bo6bo13bo38bo$o33bo20bo4bo14bo38bo$o33bo21bo3bo14bo38bo$o33bo22b3o15bo38bo$o33bo23b2o15bo38bo$o33bo22bobo15bo37bo$o33bo21bo3b2o13bo36b2o$o33bo20bo6bo12bo36bo$o33b2o18bo8b2o10bo35bo$o33b2o17b2o10b3o7bo35bo$o34bo15b2o14b9o34bo$o35bo12b2o17b2o6bo32bo$o35b2o9b2o21b2o4bo30b2o$o36b2o5b3o24b3o3bo27b2o$o35bo2b5o30b5o24b2o$o35bo4bo35b5o14b7o$bo34bo2b2o37bo3b15o$bo35b2o39bo$bo31b5o40bo$2bo28b2o4bo41bo$2bo26b3o5b2o40bo$2bo22b4o9bo41bo$3bo17b5o12bo41bo$3b2o16bo16bo42bo$4b2o14bo18bo41bo$6b2o9b3o19bo41b2o$8b10o21bo42bo$39bo42bo$39bo42bo$39bo42bo$39bo42bo$39b2o41bo$40bo41bo$40bo41bo$40bo41bo$40bo41bo$41bo40bo$41bo40bo$42bo39bo$42bo39bo$43bo37b2o$43bo37bo$44bo36bo$44bo35bo$45bo34bo$45bo34bo$46bo32b2o$46bo32bo$47bo30b2o$47b2o28b2o$48b2o24b4o$50b2o15b8o$52b15o! Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace! muzik Posts: 3301 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### 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? x = 11, y = 5, rule = B2ck3-ij5n78/S01e2ei3-k5ai8b2o$2o3b2o$bo4bo3bo$2bo2b2o$o7bo! (Check Gen 2) toroidalet Posts: 946 Joined: August 7th, 2016, 1:48 pm Location: my computer ### 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! muzik Posts: 3301 Joined: January 28th, 2016, 2:47 pm Location: Scotland ### Re: Thread for Non-CA Academic Questions The closest think I could think of was a Spirograph. x = 81, y = 96, rule = LifeHistory58.2A$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!
Gamedziner

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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!
muzik

Posts: 3301
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

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!

calcyman

Posts: 2019
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.

Probably not, as calcyman said, but none that I could find up to 10^8.
Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule (someone please search the rules)
- Find a C/10 in JustFriends
- Find a C/10 in Day and Night
AforAmpere

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How are all the bits converted in to music?
One big dirty Oro. Yeeeeeeeeee...

gameoflifemaniac

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