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Re: Thread for Non-CA Academic Questions
Posted: October 6th, 2017, 4:21 pm
by gameoflifemaniac
What's the definition of trivial?
Re: Thread for Non-CA Academic Questions
Posted: October 6th, 2017, 6:05 pm
by Apple Bottom
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.
Re: Thread for Non-CA Academic Questions
Posted: October 7th, 2017, 9:09 am
by Gamedziner
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?
Re: Thread for Non-CA Academic Questions
Posted: October 10th, 2017, 4:59 pm
by praosylen
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.
Re: Thread for Non-CA Academic Questions
Posted: October 10th, 2017, 5:13 pm
by Macbi
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)?
Re: Thread for Non-CA Academic Questions
Posted: October 10th, 2017, 5:31 pm
by praosylen
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).
Re: Thread for Non-CA Academic Questions
Posted: October 10th, 2017, 6:18 pm
by Apple Bottom
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.
Re: Thread for Non-CA Academic Questions
Posted: October 11th, 2017, 5:28 am
by Saka
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?
Re: Thread for Non-CA Academic Questions
Posted: October 11th, 2017, 6:13 am
by Bullet51
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
Re: Thread for Non-CA Academic Questions
Posted: October 11th, 2017, 6:16 am
by Macbi
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.
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 11:55 am
by muzik
Is it possible for star polygons, such as {3/2} and {5/3}, to exist on the hyperbolic plane?
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 12:51 pm
by muzik
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.
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 2:48 pm
by apg
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.
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 3:34 pm
by muzik
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?
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 4:33 pm
by Rocknlol
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.
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 4:45 pm
by muzik
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
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 5:32 pm
by apg
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
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 5:43 pm
by muzik
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!
Re: Thread for Non-CA Academic Questions
Posted: November 3rd, 2017, 9:52 pm
by toroidalet
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?
Re: Thread for Non-CA Academic Questions
Posted: November 4th, 2017, 6:12 am
by muzik
Seems as though "trefoil knot" fits this shape well enough. Can't seem to find any shapes that match the 5/3 shape though.
Re: Thread for Non-CA Academic Questions
Posted: November 4th, 2017, 7:34 am
by Gamedziner
The closest think I could think of was a Spirograph.
Re: Thread for Non-CA Academic Questions
Posted: November 6th, 2017, 4:45 pm
by muzik
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.
Re: Thread for Non-CA Academic Questions
Posted: November 6th, 2017, 5:37 pm
by apg
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.
Re: Thread for Non-CA Academic Questions
Posted: November 6th, 2017, 6:06 pm
by AforAmpere
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.
Re: Thread for Non-CA Academic Questions
Posted: November 12th, 2017, 3:40 am
by gameoflifemaniac
How are all the bits converted in to music?