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Old 2021-04-09, 21:03   #4269
Xyzzy
 
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https://www.quantamagazine.org/mathe...ture-20210405/
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Old 2021-04-15, 11:15   #4270
retina
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Default The eyes have it

When: ⅈ = √-1

Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ

And ⅈ^ⅈ^ⅈ^ⅈ^...^ⅈ^ⅈ converges to ~ 0.438282... + 0.360592...ⅈ
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Old 2021-04-15, 13:13   #4271
Dr Sardonicus
 
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Quote:
Originally Posted by retina View Post
When: ⅈ = √-1

Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ

And ⅈ^ⅈ^ⅈ^ⅈ^...^ⅈ^ⅈ converges to ~ 0.438282... + 0.360592...ⅈ
Good one! There are two different sequences here. The first one repeats five four terms cyclically, the second is the one that converges to the value indicated:
Code:
? z=I;for(i=1,20,print(i" "z);z=z^I)
1 I
2 0.20787957635076190854695561983497877003 + 0.E-39*I
3 0.E-38 - 1.0000000000000000000000000000000000000*I
4 4.8104773809653516554730356667038331263 + 0.E-57*I
5 0.E-57 + 1.0000000000000000000000000000000000000*I
6 0.20787957635076190854695561983497877003 + 0.E-58*I
7 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
8 4.8104773809653516554730356667038331263 + 0.E-57*I
9 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
10 0.20787957635076190854695561983497877003 + 0.E-58*I
11 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
12 4.8104773809653516554730356667038331263 + 0.E-57*I
13 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
14 0.20787957635076190854695561983497877003 + 0.E-58*I
15 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
16 4.8104773809653516554730356667038331263 + 0.E-57*I
17 3.186183822264904554 E-58 + 1.0000000000000000000000000000000000000*I
18 0.20787957635076190854695561983497877003 + 0.E-58*I
19 3.186183822264904554 E-58 - 1.0000000000000000000000000000000000000*I
20 4.8104773809653516554730356667038331263 + 0.E-57*I
? z=I;for(i=1,20,print(i" "z);z=I^z)
1 I
2 0.20787957635076190854695561983497877003 + 0.E-39*I
3 0.94715899807237838065347535201819333350 + 0.32076444997930853466011684587486314010*I
4 0.050092236109321075325057857667404946465 + 0.60211652703600378785675053054092402757*I
5 0.38716618108611499414398349627185887944 + 0.030527081605484415140640453871444887915*I
6 0.78227568243395340528654343812104529543 + 0.54460655765798973477763897335697905935*I
7 0.14256182316366677621483372940966688336 + 0.40046652533708730698492897511841708916*I
8 0.51978639640785427894557902248600215208 + 0.11838419641581427825928144766679494853*I
9 0.56858861727189698753818358443166667583 + 0.60507840679780382839423641672697902899*I
10 0.24236524682521112389090823201045302323 + 0.30115059207131774722222823190168820179*I
11 0.57848868337710091563881022955895887525 + 0.23152973530683357064541382292107588487*I
12 0.42734013269162773482179644068826711761 + 0.54823121734356120368611146605201176309*I
13 0.33096710435764648494764410889019260596 + 0.26289184279459072161601256135202806100*I
14 0.57427101539042836742244120057133428331 + 0.32871562363049941282674521086088084983*I
15 0.36994804215669755616736899867123840508 + 0.46817319337186955733969232674886127136*I
16 0.40063349486673663827824023315392161379 + 0.26312021335073463213299399846450098581*I
17 0.53474389203394446098262230758571465046 + 0.38932777320075335892804042289024294020*I
18 0.36211373777488820944581031376300750280 + 0.40396445699053973513867483335988567832*I
19 0.44669687505662957512158203846002241697 + 0.28556732839726256985973488224519152118*I
20 0.48769614755247653922390493148645597016 + 0.41217486803024428049635565128868599910*I

Last fiddled with by Dr Sardonicus on 2021-04-15 at 15:38 Reason: I can't count
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Old 2021-04-15, 16:12   #4272
retina
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Quote:
Originally Posted by retina View Post
When: ⅈ = √-1

Then: (((ⅈ^ⅈ)^ⅈ)^ⅈ)^ⅈ = ⅈ
Works for any x

(((x^ⅈ)^ⅈ)^ⅈ)^ⅈ = x

Last fiddled with by retina on 2021-04-15 at 16:15 Reason: Except for x == 0, of course
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Old 2021-04-16, 01:36   #4273
Dr Sardonicus
 
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Quote:
Originally Posted by retina View Post
Works for any x

(((x^ⅈ)^ⅈ)^ⅈ)^ⅈ = x
Blindly following the laws of exponents, yes, repeatedly raising to the i power you get x, xi, x-1, x-i, x, ...

Unfortunately, the "laws of exponents" can go haywire when the exponents aren't real numbers.

Example:

Code:
? z=100;for(i=0,4,print(i" "z);z=z^I)
0 100
1 -0.10701348355877020772086517528518239458 - 0.99425756941378968736161937190915602113*I
2 5.3549165552476473650304932958904718146 - 5.365580600568365703 E-41*I
3 -0.10701348355877020772086517528518239458 + 0.99425756941378968736161937190915602113*I
4 0.18674427317079888144302129348270303935 + 1.8711616494015361548219666197978930722 E-42*I
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Old 2021-04-16, 03:46   #4274
retina
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Quote:
Originally Posted by Dr Sardonicus View Post
Blindly following the laws of exponents, yes, repeatedly raising to the i power you get x, xi, x-1, x-i, x, ...

Unfortunately, the "laws of exponents" can go haywire when the exponents aren't real numbers.
Yes, you are correct. I was blind.

-e^pi < ||x|| <= e^pi
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Old 2021-04-16, 13:34   #4275
Xyzzy
 
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https://www.bbc.com/news/science-environment-56749105
Quote:
Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".
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Old 2021-04-16, 14:13   #4276
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Quote:
Originally Posted by Xyzzy View Post
Quote:
Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".
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Old 2021-04-16, 18:26   #4277
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Quote:
Originally Posted by Xyzzy View Post
Scientists in the US have developed a paint significantly "whiter than the whitest paint currently available".
That makes it too politically incorrect to use.
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Old 2021-04-16, 20:30   #4278
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Quote:
Originally Posted by Xyzzy View Post
Outstanding! For WHAM or WUPPE development, best NASA-approved paint we could use was solar spectrum absorptance ~23.%. More recently, there's 9% http://www.aztechnology.com/products...azw-la-ii.html.
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Old 2021-04-19, 00:06   #4279
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https://www.youtube.com/watch?v=8JxN33fl6bs


Quoting the video description
Quote:
Originally Posted by Youtube
We are very excited to share the first-ever camera collar footage from a wild wolf to our knowledge.

We hope you enjoy seeing the world from a wolf's point of view! What is particularly fascinating is
that this wolf (V089, a lone wolf) knew how to hunt and catch fish. He can be seen eating 3 different
fish, which were all killed and consumed at the same spot along the Ash River. Based on the
amount of time this wolf spent in this spot, it is clear this wolf killed more than 3 fish.


However, the collar only took videos for 30 seconds at the beginning of every hour of daylight meaning
we only got 7 minutes of video footage each day (14 hr of daylight x 30 second per hour). 7 minutes
of footage a day is not that much. Luckily, we still captured some really neat stuff! Up to this point,
we had only documented wolves from a single pack (the Bowman Bay Pack) hunting and killing fish
at the same small creek.


However, this footage clearly demonstrates that other wolves in our area know how to hunt fish and they
do so in different areas. This revelation—in addition to some other info we learned in 2020
(i.e., we had another wolf from the Paradise Pack that went fishing…more about this soon!)—provides insight
into the genesis and persistence of unique predation behaviors in wolf populations!


We used a Vectronic-Aerospace camera collar for this footage. The collar worked great and we
are excited to deploy more of these camera collars in the future!

Last fiddled with by firejuggler on 2021-04-19 at 00:15 Reason: formatting the quote
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