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R-pentomino neutronium

For discussion of other cellular automata.
turnip
Posts: 2
Joined: March 10th, 2014, 9:06 am

R-pentomino neutronium

I created a rule I called Neutronium, which is CGOL plus a third state, which is created when a live cell is surrounded by 8 live cells. Rather than dying, it turns into neutronium, which is considered live, and never dies:

Code: Select all

@RULE Neutronium
@TABLE
n_states:3
neighborhood:Moore
symmetries:permute
var a={1,2}
var b={1,2}
var c={1,2}
var d={1,2}
var e={1,2}
var f={1,2}
var g={1,2}
var h={1,2}

0,a,b,c,0,0,0,0,0,1
1,0,0,0,0,0,0,0,0,0
1,a,0,0,0,0,0,0,0,0
1,a,b,c,d,0,0,0,0,0
1,a,b,c,d,e,0,0,0,0
1,a,b,c,d,e,f,0,0,0
1,a,b,c,d,e,f,g,0,0
1,a,b,c,d,e,f,g,h,2
@COLORS
0 48 48 48
1 255 255 255
2 255 0 0

I ran all the pentominos through it. Most of them (not surprisingly) behaved exactly as they do in CGOL, with the exception of the I pentomino, which turns into the following in two generations:

Code: Select all

.....O
...O...O
.O...N...O
...O...O
.....O

But the oddest surprise was the r-pentonimo, which after 76,398 generations had a population of 86,158,122, including an estimated 20,000 gliders flying out in four huge sparse triangular formations (and about 20 orthogonal spaceships), fleeing a central seething ball of neutronioum growing at a rate of about 2/11c:

(this is my first post on this board. Please Let me know if I am in breach of any forum etiquette)
--Mike Turniansky

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

You're not in any breach of forum etiquette! Welcome to the forums!
This rule is interesting- and adding a live cell that never dies might make some useful signal circuitry like reflectors. It might be even more useful if Neutronium ALMOST never died. I'll look into this rule and see what I can find

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

A bit of stuff was already technically explored with Neutronium by Extrementhusiast, who made an extended life rule with Neutronium (however, it didn't occur naturally and wasn't given such a cool name. Check http://code.google.com/p/ruletablerepos ... i/TheRules)
The table was

Code: Select all

@RULE extendedlife

@TABLE

n_states:7
neighborhood:Moore
symmetries:permute

# General Behavior:
#
# 0 is empty
# 1 is on
# 2 is birthforcer
# 3 is deathforcer
# 4 is birthforcer + deathforcer
# 5 is blocker
# 6 is reactor
#
# Specific Behavior
# 0 changes to 1 when (state1 neighbors + state6 neighbors) == 3
# 0 changes to 1 when (state2 neighbors + state4 neighbors) >= 1
# otherwise 0 stays 0
#
# 1 changes to 0 when (state1 neighbors + state6 neighbors) != 2, 3
# 1 changes to 0 when (state3 neighbors + state4 neighbors) >= 1
# otherwise 1 stays 1
#
# 2 always stays 2
#
# 3 always stays 3
#
# 4 always stays 4
#
# 5 always stays 5
#
# 6 always stays 6

var a={2,4}
var b={0,1,2,3,4,5,6}
var c={b}
var d={b}
var e={b}
var f={b}
var g={b}
var h={b}
var i={1,6}
var j={i}
var k={i}
var L={i}
var m={i}
var n={i}
var o={i}
var p={i}
var q={0,2,3,4,5}
var r={q}
var s={q}
var t={q}
var u={q}
var v={q}
var w={q}
var x={q}
var y={3,4}

# 0 birthforcer
0,a,b,c,d,e,f,g,h,1

# 0 normal birth
0,i,j,k,q,r,s,t,u,1

# 1 deathforcer
1,y,b,c,d,e,f,g,h,0

# 1 normal death
1,q,r,s,t,u,v,w,x,0
1,i,r,s,t,u,v,w,x,0
1,i,j,k,L,u,v,w,x,0
1,i,j,k,L,m,v,w,x,0
1,i,j,k,L,m,n,w,x,0
1,i,j,k,L,m,n,o,x,0
1,i,j,k,L,m,n,o,p,0

@COLORS

0   0   0   0
1 255 255   0
2   0 255   0
3   0   0 255
4   0 255 255
5 153 153 153
6 255 255 128

A new eater given there was

Code: Select all

x = 66, y = 8, rule = extendedlife
56.A$54.A.A$55.2A$2.2A15.A$A4.A11.A3.A12.A2.A$6.A15.A15.A22.2A.F$A5.A
10.A4.A11.A3.A21.A2.2FE$.6A11.5A12.4A22.2A.F!  So, I made a stable form with Neutronium: Code: Select all x = 52, y = 8, rule = Neutronium 39.A$37.A.A$38.2A$2.A$A3.A12.A2.A$5.A15.A22.2A.B.2B$A4.A11.A3.A21.A2. 2B.B.B$.5A12.4A22.2A.B.2B!

Last to note, the HWSS is now unstable because a cell of Neutronium is created.

I wonder if there are any Neutronium guns, like

Code: Select all

x = 3, y = 3, rule = extendedlife
B.B2$B.B!  from extendedlife. --- P.S. stabilized the eater in a smaller form, at least for SSes: Code: Select all x = 24, y = 5, rule = Neutronium A2.A19.B$4.A13.2A.B$A3.A12.A2.3B$.4A13.2A.B$23.B!  turnip Posts: 2 Joined: March 10th, 2014, 9:06 am Re: R-pentomino neutronium Interesting to note about the the HWSS. No wonder I haven't seen any show up with the the r-pentomino pattern. BTW, when I originally started this idea, I didn't notice the limitation on how variables worked in the @TABLE, and I accidentally forgot the true isolation death rule, so my rule looked like this: Code: Select all @RULE BadNeutronium @TABLE n_states:3 neighborhood:Moore symmetries:permute var a={1,2} 0,a,a,a,0,0,0,0,0,1 1,a,0,0,0,0,0,0,0,0 1,a,a,a,a,0,0,0,0,0 1,a,a,a,a,a,0,0,0,0 1,a,a,a,a,a,a,0,0,0 1,a,a,a,a,a,a,a,0,0 1,a,a,a,a,a,a,a,a,2  Which grows an incredibly intricate and beautiful fractal-like pattern when you start with the r-pentomino. twinb7 Posts: 175 Joined: February 11th, 2014, 8:08 pm Location: Ames, Iowa Re: R-pentomino neutronium turnip wrote:Interesting to note about the the HWSS. No wonder I haven't seen any show up with the the r-pentomino pattern. BTW, when I originally started this idea, I didn't notice the limitation on how variables worked in the @TABLE, and I accidentally forgot the true isolation death rule, so my rule looked like this... So it's like B3/S023, with Neutronium? I actually like it, note this wickstretcher which occurs in the R-pentomino's evolution: Code: Select all x = 34, y = 6, rule = BadNeutronium 2.A4.A$.2A4.A2.B2.A6.B2.A6.B2.A$2A.A3.2A$.A.A5.A5.B2.A6.B2.A$2.2A$6.A
.A!

The fractal-ness is really cool, and looks almost identical to a Julia set! Namely its because the B-heptomino is a chaotic replicator.

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

Also, since Neutronium doesn't die it can easily replace the rotor of any oscillator.

Code: Select all

x = 13, y = 13, rule = Neutronium
2.ABA3.ABA2$A4.A.A4.A$B4.B.B4.B$A4.B.B4.A$2.A2B3.2BA2$2.A2B3.2BA$A4.B
.B4.A$B4.B.B4.B$A4.A.A4.A2$2.ABA3.ABA!  Other random patterns that may or may not be of interest: Code: Select all x = 4, y = 7, rule = Neutronium 2A$A.A$2.A$B.2A$2.A$A.A$2A!  Code: Select all x = 4, y = 10, rule = Neutronium 2A$A.A$2.A$B.2A$2.A$2.A$B.2A$2.A$A.A$2A!

Very small pipsquirter:

Code: Select all

x = 4, y = 5, rule = Neutronium
.2A$.B.A$3B$.B.A$.2A!

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

This rule is really cool. The following shows that you can use a block to make a very small eater with almost identical functionality to the eater 1, along with another eater that regenerates a beehive. There are also a load of neutronium core oscillators that make sparks that could be incredibly useful.

Code: Select all

x = 107, y = 73, rule = Neutronium
97.B$97.B$96.3B$97.B$96.A.A$89.B6.A.A6.B$89.B7.A7.B$88.3B13.3B$89.B
15.B$79.A8.A.A13.A.A$78.A9.A.A13.A.A$78.3A8.A15.A10$26.A$25.A7.5A11. 4A$25.3A5.A4.A10.A3.A$33.A15.A$22.BA10.A3.A11.A2.A$22.BA12.A47.A$71.A
2.A7.A3.A$58.B.2A8.A10.A$56.4B2.A7.A3.A6.A4.A$58.B.2A8.4A7.5A7$17.3A
19.B$15.2A.B.2A15.B.B.B$14.3A.B.3A13.7B$13.A.B2.B2.B.A13.B.B.B$12.A.
9B.A14.B$12.A.A.A.B.A.A.A$13.A9.A5$97.A$96.A.A$95.A3.A$2.2A7.2A83.3A$.A2.A7.A12.A$A.B.A4.3A13.A68.2A3.2A$.A.A6.B81.A3.A.A3.A$2.A6.3B11.A.B
.A11.B51.2A3.A.A3.2A$10.B12.A3BA10.3B49.A5.A.A5.A$10.B14.B13.B51.A.2A
.A.A.2A.A11$64.3A$64.A2.A$64.A$64.A$65.A.A!  Kazyan Posts: 947 Joined: February 6th, 2014, 11:02 pm Re: R-pentomino neutronium Some hand-found oscillators. Neutronium seems conductive to p3, in particular. Code: Select all x = 64, y = 16, rule = Neutronium 4$45.B$32.A$31.A.A25.2A$.A7.A6.A6.3B5.2B25.4B$.A2B5.8B6.B.B6.B5.2A.A
7.A.2A4.AB2.BA$.A2B4.2A6.2A5.3B6.B6.4B2.B2.4B5.AB2.BA$2.3A25.A.2B4.2A
.A7.A.2A5.4B$31.A.A25.2A$32.A2$45.B!  EDIT: A p8 with a strong spark. Code: Select all x = 8, y = 4, rule = Neutronium 5.A.A$2A3.2BA$2A2.A$5.B!

Last edited by Kazyan on March 11th, 2014, 8:14 pm, edited 1 time in total.
Tanner Jacobi

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

Two eaters hassling each other...

Code: Select all

x = 12, y = 3, rule = Neutronium
2.B6.B$4B4.4B$2.B6.B!

At some point I'll get to using one of those small pipsquirters to really reduce the size of reflectors.

dvgrn
Moderator
Posts: 6467
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI
Contact:

Re: R-pentomino neutronium

twinb7 wrote:Two eaters hassling each other...
Hmm, there's a p13 version too:

Code: Select all

x = 12, y = 6, rule = Neutronium
2.BA$4B$2.BA$8.AB$8.4B$8.AB! twinb7 wrote:At some point I'll get to using one of those small pipsquirters to really reduce the size of reflectors. I suppose that could work, as long as you still have workable logic circuitry and guns and such, to make reflectable gliders with. I'd be out of my element, personally, since Neutronium breaks Herschel circuitry, or nearly all of it anyway: there might just barely be a universal set left, since Fx77 and R64 are still intact. But anything with an R-pentomino stage is hopeless. Could perfectly well build new Herschel conduits out of neutronium, of course, but that's a lot of work... Pentadecathlons are toast, and so are p46 guns and reflectors. But it looks like a fair fraction of p30 logic circuitry would still be usable: Code: Select all x = 272, y = 176, rule = Neutronium 161.2A$162.A$162.A.A8.2A$163.2A6.A2.A$170.A7.5A$170.A6.A5.A$170.A7.2A 3.A$171.A2.A7.A$173.2A9$80.2A$80.3A$66.A15.2A.A$64.A.A4.3A8.A2.A5.2A$
63.A.A16.2A.A5.2A$57.2A3.A2.A7.A2.2A2.3A$57.2A4.A.A7.A3.A2.2A$64.A.A 6.A2.A$66.A8.2A4$84.A$82.A.A$83.2A$126.A9.2A$125.A.A7.4A$118.2A4.A.2A
6.A.A.A2.A.A$118.2A3.2A.2A7.2A.2A.A2.A$91.A32.A.2A7.2A.A5.2A6.2A$92.A 32.A.A8.2A4.A3.2A4.2A$90.3A33.A7.3A7.2A$136.A4.A2.A$134.A6.A.A$135.2A$235.2A$234.3A$99.A131.A.2A15.A$97.A.A124.2A5.A2.A8.3A4.A.A$98.2A29.
2A93.2A5.A.2A16.A.A$130.2A102.3A2.2A2.A7.A2.A3.2A$129.A105.2A2.A3.A7.
A.A4.2A$240.A2.A6.A.A$240.2A8.A$106.A110.A2.A$107.A93.4A11.A14.4A$105.3A14.A78.A3.A10.A3.A10.A3.A$122.2A77.A14.4A11.A$121.A.A74.2A2.A2. A26.A2.A$197.A.A$197.A46.2A$196.2A45.2A$114.A130.A$112.A.A6.2A$113.2A 6.A2.A108.2A$116.2A7.A6.2A100.2A$118.A6.A6.2A99.A18.2A9.A$125.A125.4A
7.A.A$113.2A6.A2.A121.A.A2.A2.3A5.2A.A4.2A$112.A.A6.2A96.A25.A2.A2.2A
9.2A.2A3.2A$112.A103.4A3.A.2A17.2A9.A6.2A.A$111.2A95.2A5.4A4.A.A.2A2.
2A9.2A3.A8.A5.A.A$208.2A5.A2.A3.2A.A.2A2.A.A10.2A10.A6.A$215.4A3.2A8.
3A10.A2.A$216.4A3.2A8.3A10.A.A$219.A12.3A$231.A.A$231.2A10$20.2A$20.
2A5$20.A$20.A$19.A.A$18.2A.2A103.A$17.A5.A101.2A$20.A93.2A8.2A11.2A$17.2A3.2A90.2A7.3A7.A3.3A$124.2A5.4A4.2A.A$125.2A2.A4.A4.A2.A5.2A$
126.A2.A3.A5.2A.A5.2A$108.A.A19.A.2A3.3A$106.A3.A26.2A$98.A7.A12.2A$
17.2A78.4A4.A4.A8.2A$18.A77.2A.A.A4.A16.A.A$15.3A67.2A8.3A.A2.A3.A3.A
12.2A$15.A69.2A9.2A.A.A6.A.A13.A$97.4A$98.A$109.A$107.A.A$108.2A7.A$115.2A$116.2A27.A$112.2A31.A.A$112.A.A30.2A$114.A$114.2A$64.A$62.3A$61.A$61.2A49.A$112.3A$115.A$114.2A8.A$56.2A64.A.A$57.2A64.2A$56.A62.A
$118.2A$118.A.A6$126.2A5.2A$125.2A6.A$127.A6.3A$136.A$99.A$34.2A5.2A
54.3A$34.A7.2A52.A$25.2A5.A.A6.A54.2A36.A$24.A.A5.2A99.2A$9.A.A11.A
109.A.A$4.A4.A2.A10.A2.A117.A$5.2A5.2A9.A67.3A49.A.A$2A8.A3.2A8.A.A 66.A38.A8.2A3.A9.2A$2A10.2A11.2A65.A39.4A5.2A3.A9.2A$9.A2.A8.A100.2A 9.4A4.2A3.A$9.A.A10.2A98.2A9.A2.A6.A.A$21.2A104.A5.4A7.A$127.A4.4A$84.2A46.A$83.A.A$26.2A57.A$27.2A$26.A$15.2A$14.A3.A$3.2A8.A5.A7.A$3. 2A8.A3.A.2A4.A.A48.3A$13.A5.A3.2A12.2A39.A$14.A3.A4.2A12.2A38.A$15.2A
6.2A$25.A.A$27.A$58.A8.A2.A$57.A.A10.A$50.2A3.2A3.A5.2A3.A2.A$50.2A3.
2A3.A4.A.A.2A3.4A$55.2A3.A14.4A5.2A$57.A.A5.2A8.A2.A5.2A$58.A16.4A$
74.4A$74.A! c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm Re: R-pentomino neutronium twinb7 wrote:It might be even more useful if Neutronium ALMOST never died. I did exactly that, making a switchable version of Neutronium which I named with a random stroke across my keyboard: Code: Select all @RULE Nde @TABLE n_states:3 neighborhood:Moore symmetries:permute var a={0,1,2} var b={a} var c={a} var d={a} var e={a} var f={a} var g={a} var h={a} var i={a} var j={a} var k={a} var l={a} var m={1,2} var n={m} var o={m} var p={m} var q={m} var r={m} 1,1,1,1,1,1,1,1,1,2 # a living cell with 8 neighbors becomes Neutronium 0,m,n,o,0,0,0,0,0,1 1,a,0,0,0,0,0,0,0,0 1,m,n,o,p,b,c,d,e,0 2,m,n,o,p,q,r,b,c,0 # a Neutronium cell with 6 neighbors dies @COLORS # colors from # http://necsi.org/postdocs/sayama/sdsr/java/loops.java # Color.black,Color.blue,Color.red,Color.green, # Color.yellow,Color.magenta,Color.white,Color.cyan,Color.orange 1 0 0 255 2 255 0 0  In this rule, the B-heptomino turns into a knightpuffer which releases gliders. twinb7 Posts: 175 Joined: February 11th, 2014, 8:08 pm Location: Ames, Iowa Re: R-pentomino neutronium c0b0p0 wrote: twinb7 wrote:It might be even more useful if Neutronium ALMOST never died. I did exactly that, making a switchable version of Neutronium which I named with a random stroke across my keyboard: Code: Select all @RULE Nde @TABLE n_states:3 neighborhood:Moore symmetries:permute var a={0,1,2} var b={a} var c={a} var d={a} var e={a} var f={a} var g={a} var h={a} var i={a} var j={a} var k={a} var l={a} var m={1,2} var n={m} var o={m} var p={m} var q={m} var r={m} 1,1,1,1,1,1,1,1,1,2 # a living cell with 8 neighbors becomes Neutronium 0,m,n,o,0,0,0,0,0,1 1,a,0,0,0,0,0,0,0,0 1,m,n,o,p,b,c,d,e,0 2,m,n,o,p,q,r,b,c,0 # a Neutronium cell with 6 neighbors dies @COLORS # colors from # http://necsi.org/postdocs/sayama/sdsr/java/loops.java # Color.black,Color.blue,Color.red,Color.green, # Color.yellow,Color.magenta,Color.white,Color.cyan,Color.orange 1 255 255 255 2 255 0 0  In this rule, the B-heptomino turns into a knightpuffer which releases gliders. The random stroke was nice, but I'm gonna call it Neutronium-6, because that makes it sound science-y! And I'm gonna stick with the white-red colors, but no one else needs to know about that. But alas, the wonderful, wonderful eater is unstable: Code: Select all x = 4, y = 3, rule = Nde 2.B$4B$2.B!  I suppose this one is still stable: Code: Select all x = 19, y = 7, rule = Nde 5.A$4.A10.A2.A$4.3A7.A$14.A3.A$14.4A$BA$BA!  The Herschel's evolution is really cool, sprouting a ton of B-heptomino puffers (of course, after looking at more patterns, apparently this isn't uncommon) Code: Select all x = 3, y = 4, rule = Nde A$3A$A.A$2.A!

Lastly, after a while, I realize your 6 neighbors means 'Life' neighbors, and not including Neutronium neighbors, because the center Neutronium cell here dies:

Code: Select all

x = 9, y = 3, rule = Nde
2B5.2B$.7B$2B5.2B!

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

I've been bored.

Code: Select all

x = 15, y = 11, rule = Neutronium
8.B$3.B4.3B$2.B.B6.B$2.2B.B.B.B.B$3.B.B.B.B.3B$2.2B2.4B2.B.B$2.B.2B4.
3B.B$.2B.B.3B$B2.B.2B5.B$.B2.B$5.2B!

twinb7
Posts: 175
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: R-pentomino neutronium

Useless accidental find, B-heps making a row of eaters before one is annihilated....

Code: Select all

x = 8, y = 143, rule = Nde
6.2A$5.2A$6.2A$7.A136$2.A$.2A$2A$.2A!  c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm Re: R-pentomino neutronium [quote="twinb7] Lastly, after a while, I realize your 6 neighbors are 'Life' neighbors, and not including Neutronium neighbors, because the center Neutronium cell here dies: Code: Select all x = 9, y = 3, rule = Nde 2B5.2B$.7B$2B5.2B!  [/quote] Yes, the Neutronium dies if it has six or more live neighbors, including Neutronium neighbors, due to the final line in the rule table. Below is a "chaos puffer". You will probably recognize it as Puffer 2. Code: Select all x = 5, y = 18, rule = Nde 3.A$4.A$A3.A$.4A4$A$.AA$..A$..A$.A3$3.A$4.A$A3.A$.4A!  twinb7 Posts: 175 Joined: February 11th, 2014, 8:08 pm Location: Ames, Iowa Re: R-pentomino neutronium I've been trying to make a CLEAN quadratic growth pattern with B-puffers in Nde. No success. c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm Re: R-pentomino neutronium twinb7 wrote:I've been trying to make a CLEAN quadratic growth pattern with B-puffers in Nde. No success. You would probably need rakes with a period around 520. To do that with only the spacerakes working would make the first task filtering out most of the gliders. Extrementhusiast Posts: 1824 Joined: June 16th, 2009, 11:24 pm Location: USA Re: R-pentomino neutronium twinb7 wrote:It might be even more useful if Neutronium ALMOST never died. I actually have eight variations on this; see thread "Conway's Game of Stones" for the first. I Like My Heisenburps! (and others) Alexey_Nigin Posts: 323 Joined: August 4th, 2014, 12:33 pm Location: Ann Arbor, MI Contact: Re: R-pentomino neutronium I liked Neutronium and decided to make my own variation. Code: Select all @RULE DoubleN Similar to the Neutronium rule by Mike Turniansky, but with richer behaviour. Made by Alexey Nigin (twitter.com/AlexeyNigin). @TABLE n_states:4 neighborhood:Moore symmetries:permute var a1={0,1,2,3} var a2=a1 var a3=a1 var a4=a1 var a5=a1 var a6=a1 var a7=a1 var a8=a1 var o1={1,3} var o2=o1 var o3=o1 var b1={0,2} var b2=b1 var b3=b1 var b4=b1 var b5=b1 var f1={1,2,3} var f2=f1 var f3=f1 var f4=f1 var f5=f1 var f6=f1 var f7=f1 var f8=f1 var x={0,1} 0,f1,f2,f3,f4,f5,f6,f7,f8,2 1,f1,f2,f3,f4,f5,f6,f7,f8,3 2,f1,f2,f3,f4,f5,f6,f7,f8,0 3,f1,f2,f3,f4,f5,f6,f7,f8,1 0,o1,o2,o3,b1,b2,b3,b4,b5,1 1,o1,o2,a1,b1,b2,b3,b4,b5,1 x,a1,a2,a3,a4,a5,a6,a7,a8,0 @COLORS 0 48 48 48 1 255 255 255 2 32 32 255 3 255 16 16 My rule has two almost-never-dying states. I expected it to be quite stable, but this prediction was wrong. By the way, congratulations to moderators: your headache is back on the forums, so you will not get bored in the near future. There are 10 types of people in the world: those who understand binary and those who don't. Alexey_Nigin Posts: 323 Joined: August 4th, 2014, 12:33 pm Location: Ann Arbor, MI Contact: Re: R-pentomino neutronium p19: Code: Select all x = 6, y = 5, rule = DoubleN B.A$.C2.AB$.A$2.A.C$4.B! A puffer: Code: Select all x = 23, y = 57, rule = DoubleN 3.3A11.3A$2.A17.A$3.A2.3A.3A.3A2.A$5.A2.A.A.A.A2.A$7.2A.A.A.2A$7.A2.A
.A2.A$10.A.A$7.A2.A.A2.A$8.A.A.A.A$8.A.A.A.A$5.2A.A.A.A.A.2A$5.2A.A.A
.A.A.2A$2.3A3.A.A.A.A3.3A$.A5.2A.A.A.2A5.A$2.A.2A2.A.A.A.A2.2A.A$5.2A
.A.A.A.A.2A$5.2A.A.A.A.A.2A$6.A.A.A.A.A.A$8.A.A.A.A$4.A.A.A.A.A.A.A.A
$3.3A.2A.A.A.2A.3A$3.5A2.A.A2.5A$7.2A.A.A.2A$3.2A5.A.A5.2A$2.A2.A4.A. A4.A2.A$3.A.A2.A.A.A.A2.A.A$2.2A.2A9.2A.2A$2.2A.A.A2.3A2.A.A.2A2$2.2A 3.3A3.3A3.2A$.A19.A$2A3.2A4.A4.2A3.2A$3.3A11.3A$3.A15.A$2A.A.A11.A.A.
2A$2A.A.A.2A5.2A.A.A.2A$3.A.A.2A5.2A.A.A$2.A.A.3A5.3A.A.A2$4.3A9.3A$4.3A.A.A.A.A.3A$5.5A.A.5A$10.A.A$10.3A$7.A.A3.A.A2$5.2A.3A.3A.2A$5.A 11.A$4.A2.A7.A2.A$4.A4.A3.A4.A$9.A3.A$3.6A5.6A$2.2A7.A7.2A$10.A.A$10.
3A2$11.B! Another puffer: Code: Select all x = 16, y = 22, rule = DoubleN 7.3A$6.A$9.A$4.2A$2.2A$.2A.5A$A7.A4.A$.3A2.2A.A3.2A$2.2A2.A4.A3.A$11.
2A2.A$15.A$15.A$11.2A2.A$2.2A2.A4.A3.A$.3A2.2A.A3.2A$A7.A4.A$.2A.5A$
2.2A$4.2A$9.A$6.A$7.3A!
Look at the weekender:

Code: Select all

x = 16, y = 11, rule = DoubleN
.A12.A$.A12.A$A.A10.A.A$.A12.A$.A12.A$2.A3.4A3.A$6.4A$2.4A4.4A2$4.A6.
A$5.2A2.2A! There are 10 types of people in the world: those who understand binary and those who don't. A for awesome Posts: 1993 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: Re: R-pentomino neutronium Four different glider-neutronium collisions: Code: Select all x = 31, y = 5, rule = DoubleN C10.C10.C6.C2$.3A8.2A8.2A4.2A$.A10.A.A7.A.A3.A.A$2.A9.A9.A5.A!

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

BlinkerSpawn
Posts: 1950
Joined: November 8th, 2014, 8:48 pm
Location: Getting a snacker from R-Bee's

Re: R-pentomino neutronium

p31 can reflect gliders if supported by neutronium:

Code: Select all

x = 28, y = 19, rule = DoubleN
5.A9.A.A7.A.A$5.A.A7.2A8.2A$5.2A9.A9.A2$25.B3$.B9.B8.B3$2.3A7.3A7.3A$
.A3.A5.A3.A5.A3.A$A5.A3.A5.A3.A5.A$3.A9.A9.A$2.A.A7.A.A7.A.A$BC3.CB3.
BC3.CB3.BC3.CB$.A.A.A5.A.A.A5.A.A.A$2.3C7.3C7.3C$3.B9.B9.B!  R-to-Pi-to-G, repeat time 29(!): Code: Select all x = 13, y = 11, rule = DoubleN 11.C$10.B$5.B3$2A$.2A$.A10.B3$2.B3.BCB!  H-to-G, repeat time 62: Code: Select all x = 18, y = 25, rule = DoubleN 11.B2$B3$14.B3$7.B4.C3.B3$16.B4$15.B7$15.3A$16.A$14.3A!  Beehive stopper: Code: Select all x = 9, y = 17, rule = DoubleN 2.2B$2.2C$2.2A2$.A$.2A$A.A$5.B7$7.A$6.2A$6.A.A!

Smallest found oscillators for periods 60 or less (except 47, 51, 58, and 59):

Code: Select all

x = 194, y = 115, rule = DoubleN
102.B2.B38.C$74.A2CA56.A2C5.C.C$100.2A31.A2.A6.B20.A2.A$3B3.A23.B.3B 25.3B.3B7.C2.C12.3B.3B3.ACA17.B.B.3B6.2CA14.3B.3B7.A2CA$2.B3.A23.B3.B
3.C2B2C19.B3.B25.B3.B4.2BC16.B.B3.B23.B5.B8.2A$3B3.A23.B.3B2.B2C20.3B .3B7.A2CA12.3B.3B7.B15.3B.3B23.3B.3B8.2B$B29.B.B5.A3.A17.B3.B27.B.B
27.B.B27.B.B$3B27.B.3B8.A16.3B.3B23.3B.3B25.B.3B23.3B.3B27.B3.B$39.B$180.B11.B$114.B$179.B5.B.B5.B2$113.A.A$3B27.B.3B7.A17.3B.3B23.3B.3B 17.C5.B.B.3B23.3B.3B26.2A3.2A$2.B27.B3.B5.B2.C18.B3.B10.B14.B3.B16.C.
C4.B.B3.B6.B16.B5.B25.B.C.C.C.B$3B4.C22.B.3B6.CA17.3B.3B7.BAC13.3B.3B 15.AC.CA3.3B.3B5.C2.B14.3B.3B26.AB3.BA$2.B2.C.C22.B3.B5.B2.C16.B5.B5.
B19.B3.B15.B3.B5.B3.B4.C.A18.B3.B27.C.C.C$3B3.B23.B.3B7.A17.3B.3B5.2C .A14.3B.3B25.B.3B3.B2C17.3B.3B$72.C.B56.CA$138.B$133.B3$73.B93.A2CA$B
.B4.C22.B.B.B13.A2C9.3B.B.B23.3B.B.B23.B.B.B.B23.3B.B.B10.C2.C$B.B27. B.B.B12.A2.A11.B.B.B5.ACB17.B.B.B7.B3.B11.B.B.B.B23.B3.B.B9.AC2.CA$3B
2.3C22.B.3B12.2CA10.3B.3B5.AC16.3B.3B12.CB9.3B.3B14.A2CA5.3B.3B9.A4.A
$2.B27.B3.B25.B5.B4.C.B18.B3.B11.A.C11.B3.B25.B3.B10.B2CB$2.B4.C22.B
3.B25.3B3.B3.B2C17.3B3.B7.B3.A.CB10.B3.B14.C2.C5.3B3.B$108.A.C22.2C$
109.CB22.C2.A4.A2CA$104.B3.B$134.A.C3$3B4.C22.B.3B6.2CB16.3B.3B13.B9. 3B.3B7.B15.B.B.3B23.3B.3B23.2A4.2A$B5.A23.B.B7.B2.BC17.B.B16.AC9.B.B
25.B.B.B4.B.CB17.B3.B25.2A4.2CB$3B2.C.C22.B.3B8.ACB14.3B.3B13.C9.3B. 3B5.BC16.3B.3B.A.2A18.3B.3B30.A$2.B3.A23.B3.B8.A16.B5.B16.C8.B3.B5.C.
A17.B3.B.AC22.B3.B28.B$3B4.C22.B.3B25.3B.3B14.CA7.3B.3B4.B2C18.B.3B3. A2.B16.3B.3B$83.B18.CA24.AC$128.A.B$103.B$180.CA2.AC$114.B66.B2CB$72. B$3B27.B.3B5.2C18.3B.3B4.AC17.3B.3B18.C4.B.B.3B12.B10.3B.3B3.B7.B$B4. B2.C21.B.B6.A2.A19.B.B5.C5.B15.B.B18.ACA4.B.B.B11.B6.B6.B3.B6.2A$3B2.
CACB21.B.3B4.C2.C17.3B.3B2.BC19.3B.3B16.C6.3B.3B15.AC6.3B.3B$B.B2.B2. C21.B.B.B4.A2.A17.B3.B.B3.C5.B15.B.B.B16.C8.B.B.B11.3A2.C8.B.B.B4.BCB CBCB$.2B27.B2.2B25.3B2.2B4.AC17.3B2.2B16.ACA6.B2.2B11.A2.2C7.3B2.2B$72.B42.C24.C$139.AC$114.B24.B2$101.B3.B$168.2C3.2A$3B3.B2C21.B.3B25.
3B.3B23.3B.3B5.A.A15.B.B.3B23.3B.3B10.B5.A$2.B3.CA22.B3.B27.B3.B25.B 3.B6.C16.B.B3.B23.B5.B11.C7.B$2.B27.B3.B12.B12.3B3.B4.B18.3B3.B5.C.C
15.3B3.B23.3B3.B18.C.C$2.B27.B3.B11.C13.B5.B25.B3.B4.AC.CA16.B3.B25.B 3.B18.C.C$2.B5.AC20.B3.B10.C14.3B3.B4.C2ACB14.3B3.B25.B3.B23.3B3.B19.
B$7.2CB34.B2CA2.B19.A2C.C98.A$45.2A23.A2.A99.2A$71.C62.C$134.C.C$135. B6.2C$8.B132.A2.A$3B4.ACA20.B.3B5.A19.3B.3B9.B4.A.B6.3B.3B23.B.B.3B 14.C2.C5.3B.3B$B.B5.C.A19.B.B.B4.A.A20.B.B.B13.2A10.B.B.B6.B16.B.B.B.
B14.A2.A5.B3.B.B$.B7.A20.B2.B5.A.A18.3B2.B15.A8.3B2.B5.B18.3B2.B24.3B 2.B$B.B3.BA22.B.B.B5.C19.B3.B.B25.B.B.B3.AC20.B.B.B25.B.B.B$3B27.B.3B 5.C19.3B.3B11.B2.B8.3B.3B2.C22.B.3B23.3B.3B$39.A.A56.BCA6.C28.B$40.A 58.C$100.AC$101.B$103.B36.B2$2B28.B.2B11.CB13.3B.2B7.B3.B12.3B.2B15.B C7.B.B.2B7.B16.3B.2B$B.B7.A19.B.B.B9.BC16.B.B.B25.B.B.B15.C7.B.B.B.B
5.AC16.B3.B.B$3B3.B2.2CB18.B.3B9.AC14.3B.3B6.C.A.C12.3B.3B6.A8.2C6.3B .3B4.C3.A14.3B.3B$2.B7.A19.B3.B9.A2.C12.B5.B5.BA3.AB13.B3.B4.B2.C7.2C
8.B3.B3.BC3.A3.C12.B3.B$3B27.B.3B11.B13.3B.3B7.C.C13.3B.3B5.CA8.C9.B. 3B4.C3.A14.3B.3B$74.B.B24.B2.C6.BC19.AC$75.C27.A29.B2$140.B$112.CA$
55.C55.BC48.C$B.3B6.3A16.3B.3B18.B4.3B.3B3.2CBC16.B.B.3B14.C2A36.3B. 3B4.C.C$B.B.B5.CB20.B.B.B17.AC6.B.B.B2.B3.CB15.B.B.B.B7.A31.B13.B3.B.
B5.B$B.B.B6.CA17.3B.B.B18.B4.3B.B.B5.ACB15.3B.B.B6.A.A44.3B.B.B$B.B.B
25.B3.B.B17.AC6.B.B.B5.A19.B.B.B5.B2C45.B.B.B.B7.BC2A$B.3B25.3B.3B15. B2.C4.3B.3B25.B.3B6.A2.A5.2A37.2B.3B8.C2.A$54.C51.B5.2A52.3CB$133.AC 7.B24.B$134.CB6.A.C$121.3B.3B5.2AC6.C$121.B3.B.B6.A10.C$40.B30.B49.3B .B.B15.C.A$B.B6.C20.3B.B5.C19.3B.B5.AC18.B.B.B28.B.B.B17.B$B.B5.B.A 21.B.B4.B2A3.B16.B.B4.C20.B.B.B26.3B.3B5.2A$B.B5.2C20.3B.B4.CA19.3B.B
3.BCA19.3B.B6.B31.2A$B.B5.2A20.B3.B4.A22.B.B4.C22.B.B8.A$B.B27.3B.B
25.3B.B5.AC20.B.B11.B$71.B31.C.A$103.2CA$102.C.CB$121.3B.B$121.B3.B$
121.3B.B$123.B.B$121.3B.B!

(Collection last updated 8-29-2015)
This rule is fun.
Last edited by BlinkerSpawn on August 29th, 2015, 9:05 am, edited 7 times in total.
LifeWiki: Like Wikipedia but with more spaceships. [citation needed]

BlinkerSpawn
Posts: 1950
Joined: November 8th, 2014, 8:48 pm
Location: Getting a snacker from R-Bee's

Re: R-pentomino neutronium

Also, a reflector with repeat time 48 (even though the Snark works flawlessly at 43 ticks):

Code: Select all

x = 34, y = 22, rule = DoubleN
.A$2.A$3A10$13.A$14.A7.B$12.3A$18.B14.B$9.B7.C5.B2$31.B$26.B$11.B$15. 2B2.B!  Edited: And a tiny p88 Herschel factory hooked up to a smaller H-to-G: Code: Select all x = 35, y = 11, rule = DoubleN 34.B$23.B3$3.B$2.C2.B$.C31.B$B2C18.A$.C2.A2.2A.A10.A.A$3.3A.2A2.A9.3A
$3.B2A2.A2.A11.A!  LifeWiki: Like Wikipedia but with more spaceships. [citation needed] BlinkerSpawn Posts: 1950 Joined: November 8th, 2014, 8:48 pm Location: Getting a snacker from R-Bee's Re: R-pentomino neutronium Sorry for the triple post, but DoubleN is onniperiodic thanks to a new 39-tick reflector. Code: Select all x = 22, y = 21, rule = DoubleN 19.A.A$19.2A$20.A7$.C8.A$2.B6.A$4.B4.3A$6.B4$B2$6.B$4.ACB$3.BCA!  LifeWiki: Like Wikipedia but with more spaceships. [citation needed] A for awesome Posts: 1993 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: Re: R-pentomino neutronium G-to-more-accessible-pi (with attached pi track) with an abysmal repeat time of 132: Code: Select all x = 56, y = 39, rule = DoubleN 53.A.A$53.2A$54.A6$3.B3$B5.B3$B8.B3$3.B8.B3$6.B8.B3$9.B3$19.B$10.B9.B 3$21.B$17.B2$16.B3.A.A$20.2A$21.A$22.B$20.ACB\$15.B3.BCA!

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


text:
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### LaTeX Friends

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## Minipage Alignment

### LaTeX Forum: Graphics, Figures & Tables

Add tags Information and discussion about graphics, figures & tables in LaTeX documents.
I'm trying to get three minipages to align properly on my custom titlepage, the first two vertically aligned along the first line and the third one lying below the first. But it's not coming out quite right, the first two won't align along the "Mentors" line and the third minipage comes out below the third but it's indented. But if I don't put the \vspace command in, it runs right below the minipage, which I think looks awkward.

Code: Select all  •  Open in online LaTeX editor
\begin{minipage}{0.5\textwidth}\begin{flushleft}\normalsize\textbf{Industry Mentors:}\\Person #1 \\Person #2\\Person #3 \\\end{flushleft}\end{minipage}\begin{minipage}{0.5\textwidth}\begin{flushright}\normalsize \textbf{Academic Mentor}\\Person #4\end{flushright}\end{minipage}\vspace{0.5cm}\begin{minipage}[c]{0.5\textwidth}\begin{flushleft}{ \normalsize \textbf{Student Researchers:}\\Person #5 (project manager)Person #6\\Person #7\\Person #8\\\end{flushleft}\end{minipage}
mdk31

Posts: 20
Joined: Wed Mar 11th, 2009

To make the first two minipages line up, use the [t] option to align them both at the top vertically.

I wasn't exactly sure what you meant about the third one being indented. If I had to guess though, it's probably that a space is being put in because you didn't comment out the linefeed at the end of the \vspace command. Might as well put in the \noindent command as well.

Try something like:

Code: Select all  •  Open in online LaTeX editor
\documentclass{article}\begin{document}\noindent%\begin{minipage}[t]{0.5\textwidth}\begin{flushleft}\normalsize\textbf{Industry Mentors:}\\Person \#1 \\Person \#2\\Person \#3 \\\end{flushleft}\end{minipage}\begin{minipage}[t]{0.5\textwidth}\begin{flushright}\normalsize\textbf{Academic Mentor}\\Person \#4\end{flushright}\end{minipage}\vspace{0.5cm}%\noindent%\begin{minipage}[t]{0.5\textwidth}\begin{flushleft}\normalsize \textbf{Student Researchers:}\\Person \#5 (project manager)\\Person \#6\\Person \#7\\Person \#8\\\end{flushleft}\end{minipage}\end{document}

However, since you didn't provide a full, compiliable example, it may well be that there are differences between what I have here and what you had in mind. (Your code used # where it needs \#, and there's a stray { or a missing } on one of the lines.)

frabjous

Posts: 2065
Joined: Fri Mar 6th, 2009
Location: Amherst, MA

Bless you, thanks, this works perfectly! I can't tell you how many times this forum has helped me with a LaTeX problem.
mdk31

Posts: 20
Joined: Wed Mar 11th, 2009

Hi mdk31,

great to hear that it's working.
Generally if a problem has been solved please the topic as "solved". This can be done by editing the initial post changing the topic icon to the green checkmark.

Kind regards,

Stefan

Stefan_K

format: plain


text:
# Second Principle of Finite Induction/Zero-Based

## Theorem

Let $S \subseteq \N$ be a subset of the natural numbers.

Suppose that:

$(1): \quad 0 \in S$
$(2): \quad \forall n \in \N: \paren {\forall k: 0 \le k \le n \implies k \in S} \implies n + 1 \in S$

Then:

$S = \N$

## Proof

Define $T$ as:

$T = \set {n \in \N : \forall k: 0 \le k \le n: k \in S}$

Since $n \le n$, it follows that $T \subseteq S$.

Therefore, it will suffice to show that:

$\forall n \ge 0: n \in T$

Firstly, we have that $0 \in T$ if and only if the following condition holds:

$\forall k: 0 \le k \le 0 \implies k \in S$

Since $0 \in S$, it thus follows that $0 \in T$.

Now suppose that $n \in T$; that is:

$\forall k: 0 \le k \le n \implies k \in S$

By $(2)$, this implies:

$n + 1 \in S$

Thus, we have:

$\forall k: 0 \le k \le n + 1 \implies k \in S$

Therefore, $n + 1 \in T$.

Hence, by the Principle of Finite Induction:

$\forall n \ge 0: n \in T$

That is:

$T = \N$

and as $S \subseteq \N$ it follows that:

$S = N$

$\blacksquare$


text:
MathSciNet bibliographic data MR1737627 (2001c:60135) 60J80 (05C80 60E15 60G50) Benjamini, Itai; Häggström, Olle; Mossel, Elchanan On random graph homomorphisms into ${\bf Z}$${\bf Z}$. J. Combin. Theory Ser. B 78 (2000), no. 1, 86–114. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# What five letters am I thinking of? (Simplified Jotto.)

## 1. Warmup

"I'm thinking of five letters," I smiled.

"Are they ... G, B, K, J, and N?" you asked.

"No, none of those."

"C, X, Q, A, and N?"

I gulped. "Four of those."

"K, I, Y, T, and M?"

"One of those."

"S, D, Z, I, and K?"

"One of those. Now that's enough guessing; you have enough information to figure it out!"

This is of course a simplified version of the word game Jotto, with the difference that here the answers and guesses are not limited to real words. This also cuts out the occasional anagram-solving component of late-game Jotto.

You might also call it a non-word, white-peg-only version of Word Mastermind. Or an alphabetical cows-only version of Bulls and Cows.

Here are three more, in approximately increasing difficulty. I made sure they were all solvable without a computer, but have at it however you like.

## 2. Easy

"I Y L H C?" --"None."

"O H K E F?" --"One."

"U R I Q E?" --"Two."

"Y N A W R?" --"Three."

"J N H L X?" --"None."

"J Q O V C?" --"Two. That's enough!"

## 3. Less easy

"W Q P Z N?" --"One."

"V Y H A Z?" --"Three."

"B R I Q C?" --"One."

"B A T K F?" --"Two."

"V U M R J?" --"None."

"V N H F I?" --"Three."

## 4. Least easy

"V X K H S?" --"Two."

"T C I O L?" --"One."

"G V U F M?" --"None."

"L Q P F S?" --"Two."

"V Z X Q S?" --"One."

"H O Q V R?" --"Three."

"K Z O A T?" --"Two."

# 1

These puzzles can be solved by narrowing down the clues gradually. For example, let's remove all the letters in the "none" clue. This gives:

CXQA 4
IYTM 1
SDZI 1
Now C X Q A must all be chosen. There's only one letter left, so it must be the letter the last two rows have in common, which is I.

# 2

Removing the letters in a "none" gives the following:
OKEF 1
URQE 2
AWR 3
QOV 2
So A W R must be three of the letters. Removing those from other clues (and decrementing the counts appropriately) gives:

OKEF 1
UQE 1
QOV 2
And now both of the remaining letters have to be in QOV - so we can take out all letters that don't appear there.

O 1
Q 1
QOV 2
So O Q are the remaining letters.

# 3

Start by removing the letters in the "none" clue again. This gives:

WQPZN 1
VYHAZ 3
BIQC 1
BATKF 2
NHFI 3
Note the two 3 clues; the only letter they share is H. If the H is not chosen, there are at least six letters chosen total. So the H must be chosen, and the remaining letters must be in one of the two 3 clues.

Removing the unnecessary letters (and the H):

ZN 1
VYAZ 2
I 1
AF 2
NFI 2

So I must be another chosen letter. Using the earlier "can't pick too many letters" argument on rows 2 and 4, A must be chosen as well.

ZN 1
VYZ 1
F 1
NF 1
And now the remaining two letters must be F and Z.

# 4

XKHS 2
TCIOL 1
LQPS 2
ZXQS 1
HOQR 3
KZOAT 2

Say H is not chosen - that is, O, Q, and R are chosen. Then the first row implies that at least one of {X,S} is chosen; but means that we choose at least two from ZXQS. So H must be chosen.

XKS 1
TCIOL 1
LQPS 2
ZXQS 1
OQR 2
KZOAT 2
Now, LQPS and KZOAT are disjoint, and their clues are both 2. So all 4 of our remaining letters must be from those two sets, and any letter besides those can be discarded.

KS 1
TOL 1
LQPS 2
ZQS 1
OQ 2
KZOAT 2

Now O and Q must be chosen. So T/L/Z/S are not chosen, and the clues simplify to
K 1
P 1
KAT 1
So K and P are the remaining two letters.

• Thanks! So elegantly explained that it might help me code a solver program. Oct 25, 2021 at 23:17
• Nice. In the second listing from your solution to puzzle 3, you have the line AF 2, so you can jump straight to including A and F from there. Oct 26, 2021 at 3:08

Four of ACNQX, one of IKMTY; note that those two sets are disjoint, so those are all five of our letters. One of DIKSZ; the only one of those mentioned above is I, so we have that plus four of ACNQX. We have none of BGJKN; in particular we don't have N so that's the absent one from ACNQX. So our five letters are ACIQX.

None of ICHJLNXY. Three of ANRWY but neither of NY, so all of ARW. Two of CJOQV, disjoint from ARW so these are all our letters. One of EFHKO which must therefore be O. Two of EIQRU; one is R, already known; the other must be Q. So our letters are AOQRW.

Three of AHVYZ and three of FHINV, which come to more than our total number of letters, so we have H or V. We don't have any of JMRUV so in fact we have H but not V. And with an overlap of just one between AHVYZ and FHINV, these must be all our letters: H, two of AYZ, two of FIN. We have one of BCIQR, which can only be I. We have two of ABFKT, which must be AF. (So now we have our two of FIN and one from AYZ.) And we have one of NPQWZ, which must be N or Z, so the remaining letter is Z. AFHIZ.

None of FGMUV so let's remove those from the other info we have: HKSX:2 CILOT:1 LPQS:2 QSXZ:1 HOQR:3 AKOTZ:2. If we don't have H then we have KSX:2, OQR:3 but then there's no possible way to have AKOTZ:2. So we must have H. That leaves KSX:1 CILOT:1 LPQS:2 QSXZ:1 OQR:2 AKOTZ:2, and four letters to find. If we don't have O then we have QR:2 and AKTZ:2; then CILOT:1 means we have T so at this point we have HQRT+1 and KSX:1 LPS:1 AKZ:1 and, alas, those don't have a common letter; so we do after all have O. So we have HO+3, and KSX:1 CILT:0 LPQS:2 QSXZ:1 QR:1 AKTZ:1. CILT:0 means we can reduce the others to KSX:1 PQS:2 QSXZ:1 QR:1 AKZ:1. If not Q then PRS are all immediate and then AKZ:1 is impossible; so we have HOQ+2, and KSX:1 PS:1 AKZ:1; but also SXZ:0 so these reduce to K:1 P:1 AK:1, which means our last two letters are KP. So the full set is HKOPQ.

• Apparently I'm supposed to avoid comments like "+1" and "thanks"! But +1, and thanks! I think you might have beat Deusovi to the punch by a few minutes, too. Both excellent answers. Oct 25, 2021 at 23:23
• Yes, I was about five minutes quicker (or maybe started about five minutes earlier) than Deusovi. (For the avoidance of doubt, you are not required to give the checkmark to the earlier answer, and I am not suggesting that you should have..) Oct 26, 2021 at 11:19


text:
+0

# help

-1
24
1
+418

Five balls are numbered with the integers 1 through 5 and placed in a jar. Three are drawn without replacement. What is the probability that the sum of the three integers on the balls is odd? Express your answer as a common fraction.

Logic  Nov 2, 2018

### Best Answer

#1
+2729
+1

The balls can sum to odd if they are

(o, o, o), (o, e, e)

There is only 1 way to choose (o,o,o), i.e. (1,3,5)

There are 3 ways to choose (o,e,e), both the two even numbers, 2 and 4, and one of the odd numbers.

Thus there are 4 total ways 3 balls can sum to an odd number.

There are $$\dbinom{5}{3} = 10$$ total ways to select 3 balls from the 5.  Thus

$$P[\text{choose 3 balls that sum to an odd number}]=\dfrac{4}{10}=\dfrac 2 5$$

Rom  Nov 3, 2018
#1
+2729
+1
Best Answer

The balls can sum to odd if they are

(o, o, o), (o, e, e)

There is only 1 way to choose (o,o,o), i.e. (1,3,5)

There are 3 ways to choose (o,e,e), both the two even numbers, 2 and 4, and one of the odd numbers.

Thus there are 4 total ways 3 balls can sum to an odd number.

There are $$\dbinom{5}{3} = 10$$ total ways to select 3 balls from the 5.  Thus

$$P[\text{choose 3 balls that sum to an odd number}]=\dfrac{4}{10}=\dfrac 2 5$$

Rom  Nov 3, 2018

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text:
## Topological Methods in Nonlinear Analysis

### The Borsuk-Ulam property for cyclic groups

#### Abstract

An orthogonal representation $V$ of a group $G$ is said to have the Borsuk-Ulam property if the existence of an equivariant map $f:S(W) \rightarrow S(V)$ from a sphere of representation $W$ into a sphere of representation $V$ implies that $\dim W \leq \dim V$. It is known that a sufficient condition for $V$ to have the Borsuk-Ulam property is the nontriviality of its Euler class ${\text{\bf e}}(V)\in H^{*} (BG;\mathcal R)$. Our purpose is to show that ${\text{\bf e}}(V) \neq 0$ is also necessary if $G$ is a cyclic group of odd and double odd order. For a finite group $G$ with periodic cohomology an estimate for $G$-category of a $G$-space $X$ is also derived.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 16, Number 1 (2000), 65-72.

Dates
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.tmna/1471875422

Mathematical Reviews number (MathSciNet)
MR1805039

Zentralblatt MATH identifier
0998.57059

#### Citation

Izydorek, Marek; Marzantowicz, Wacław. The Borsuk-Ulam property for cyclic groups. Topol. Methods Nonlinear Anal. 16 (2000), no. 1, 65--72. https://projecteuclid.org/euclid.tmna/1471875422

#### References

• T. Bartsch, Critical orbits of invariant functionals and symmetry breaking , Manuscripta Math., 66 (1989), 129–152 \ref ––––, On the genus of representation spheres , Comment. Math. Helv., 65 (1990), 85–95 \ref ––––, Borsuk-Ulam theorems for $p$-groups and counterexamples for non $p$-groups , Heidelberg (1989). preprint ;, Topological Methods for Variational Problems with Symmetries , Lecture Notes in Math., 1560 , Springer-Verlag, Berlin (1993) \ref
• G. Bredon, Introduction to Compact Transformation Groups, Academic Press (1972) \ref
• K. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Springer-Verlag, 87 (1982) \ref
• M. Clapp and D. Puppe, Invariants of the Lusternik–Schnirelman type and the topology of critical sets , Trans. Amer. Math. Soc., 298 (1986), 603–620 \ref ––––, Critical point theory with symmetries , J. Reine Angew. Math., 418 (1991), 1–29 \ref
• E. Fadell, The equivariant Lusternik–Schnirelman method for invariant functionals and relative cohomological index theory , Topological Methods in Nonlinear Analysis, Sem. Math. Sup., 95 , Presses Univ., Montreal (1985), 41–76 \ref
• P. Hilton and S. Stammbach, A course in Homological Algebra, Springer-Verlag, Graduate Texts in Mathematics, 4 (1971) \ref
• W. Y. Hsiang, Cohomological Theory of Topological Transformation Groups, Springer-Verlag (1975) \ref
• M. Izydorek and W. Marzantowicz, Equivariant maps between cohomology spheres , Topol. Methods Nonlinear Anal., 5 (1995), 279-290 \ref
• W. Marzantowicz, A $G$-Lusternik–Schnirelman category of space with an action of a compact Lie group , Topology, 28 (1989), 403–412 \ref ––––, Borsuk–Ulam theorem for any compact Lie group , J. London Math. Soc. (2), 49 (1994), 195–208 \ref
• E. Spanier, Algebraic Topology, McGraw–Hill (1966) \ref
• R. Switzer, Algebraic Topology –- Homotopy and Homology, Springer-Verlag, Die Grundlehren der math. Wiss., Band 212 (1975) \ref
• T. tom Dieck, Transformation Groups, de Gruyter Studies in Mathematics, 8 (1987) \ref
• E. Weiss, Cohomology of Groups, Academic Press (1969)


text:
# 5.5. Sparse Norm Inversion¶

Here, we use magsen3d_60.exe to compute the sensitivity matrix required for the inversion; which is scaled by distance weighting. We then use maginv3d_60.exe to invert the data using sparse norms to recover a model compact susceptibility model. To keep the example simple, we added Gaussian noise with a standard deviation of 0.5 nT to all data points. We then assigned uncertainties of 0.5 nT to all magnetic data. In practice, the noise on the data is not trivial to quantify and choosing appropriate uncertainties is very important for successful inversion.

Files relevant to this part of the example are in the sub-folder inv_sparse . Before running this example, you may want to do the following:

## 5.5.1. Sensitivities¶

Here, the code magsen3d_60.exe and the input file sens.inp (see format ) are used to construct the sensitivity matrix and scale it using distance weighting. The distance weighting is applied to the sensitivity matrix to counteract the inversion’s natural tendancy to incorrectly place anomalous structures near the observation locations.

To compute the sensitivities, the following input file was used. Since we are no longer performed an least-squares inversion, a flag of 0 must be entered on the last line of the input file.

## 5.5.2. Inversion¶

Here, the code maginv3d_60.exe and the input file inv.inp (see format ) was used to recover a susceptibility model. You cannot perform the inversion until you have created the sensitivity matrix. For this example, we set P=0 and Qx=Qy=Qz=2. That is, we would like to recover a model that is compact but still smooth. To see how these parameters impact the recovered model, see the GIFtools cookbook .

The true model (left), recovered model using least-squares (middle) and recovered model using sparse norms (right) are shown below. Unlike the least-squares result, the sparse norm result is a compact structure whose maximum amplitude is much closer to that of the true model. And the distance weighting is able to place the center of the recovered model at the correct depth.


text:
# Tag Info

2

A prediction model that is correct $50\%$ of the time can be profitable if the model gains more when it is right than it loses when it is wrong. You could simplify it like this: A trading strategy is profitable if your trades have positive expected value. Now suppose that your gains when your model is right equals the losses when your model is wrong. If ...

1

I saw this paper by Deutsche through FT Alphaville Markets live.

1

The mean could be the long run variance which is sig2 = fit.Constant/(1-fit.GARCH{1}-fit.ARCH{1}); I hope this explains. If not, note I ran this model through Matlab, I get different values. you can paste your m1 and m2 values and some other intermediate results so I can see why Matlab differs. EDIT: The question refers to forecasting the returns. ...

1

A very reference can be found here: http://www.asiapacfinance.com/trading-strategies/technicalindicators

1

The TA_lib Technical Analysis library here has open source code for numerous indicators.

1

A stock's returns are also correlated to the performance of the ETFs or Index Funds that include that stock; for example, if you look at returns for MSFT (Microsoft), you might want to look at the performance of QQQ as well.

1

Very often a stock's return is determined primarily by what the broad market, or perhaps the stock's sector, is doing: to see this, take a random stock, and it will be very hard to justify most moves - until you consider the market [e.g. S&P500] or sector [e.g. transportation]. Thus, I would include the beta coefficient, which measures the volatility in ...

1

I will try to answer both of your questions together. First, regarding a standard for calculating price momentum I would say no, there is no universal standard. In addition to what is simply called the momentum indicator (which @chjortlund described in his answer), there are dozens of additional momentum indicators, and each puts an emphasis on ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Product of delta exponentials with fixed t and s

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Let $\mathbb{T}$ be a time scale, $t,s \in \mathbb{T}$, and let $p,q \in \mathcal{R}\left(\mathbb{T},\mathbb{C}\right)$ be regressive functions. The following formula holds: $$e_p(t,s;\mathbb{T})e_q(t,s;\mathbb{T})=e_{p \oplus q}(t,s;\mathbb{T}),$$ where $e_p$ denotes the delta exponential and $\oplus$ denotes circle plus.


text:
## Wednesday, December 31, 2014

### 359: Lines and Diagonals

There's sequences all over the place. What are they?

Does your sequence predict the 20th case?

And the 11th and 12th?

## Tuesday, December 30, 2014

### 358: Division and Remainders

Is that diagram correct?

What if the question were 13 ÷ 5 = 2 R3? Could we diagram it in the same way?

## Monday, December 29, 2014

### 357: Box Office Receipts for The Interview.

"Sony doesn't say" .... but do we have enough information to tell anyway?
‘The Interview’ Brings In $15 Million on Web LOS ANGELES — “The Interview” generated roughly$15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday.

Sony did not say how much of that total represented $6 digital rentals versus$15 sales. The studio said there were about two million transactions over all.

source. @ddmeyer

## Sunday, December 28, 2014

### 356: Partridge in a Number Tree

You know the song. "And a Partridge in a Pear tree." What patterns of numbers can we find here?

 The first partridge.

If you look at the total gifts each day, what sequence of numbers is this?

 Four Calling Birds, calling out numbers ...
How many different ways are there to find the total number of gifts given over the twelve days?

## Saturday, December 27, 2014

### 355: Millionaire

Would you guess? $250,000 if correct and$100,000 if she refuses to try for it.

## Friday, December 26, 2014

### 354: How to Teach Division

So, students. You've had a chance to weigh in on addition and multiplication.

What is the best way to do division?

## Thursday, December 25, 2014

### 353: How to Teach Multiplication

There are several ways to teach multiplication. Many people seem to feel that students know best how they learn, so I'm asking students to weigh in on this particular issue.

Which method is best? Is there a difference between what we should be doing with elementary students and with high school students? With how much and with what do students need to graduate high school and enter the RealWorld?

Hindu Lattice Method Grid Method

Standard Algorithm

Japanese Sticks

## Wednesday, December 24, 2014

### 352: How to Teach Addition

Everyone seems to have an opinion and now, students, we're asking for yours.

Scott Macleod says, "We now live within multidirectional conversation spaces in which 12-year-olds can reach audiences at scales that previously were reserved for major media companies, large corporations, and governments. We all now can have a voice. We all now can be publishers. We all now can find each other’s thoughts and ideas and can share, cooperate, collaborate, and take collective action."

So how should addition, and by extension subtraction, be taught?The standard algorithm or by piecewise addition?

Weigh in on the "Letter to Jack".

How would you teach these two problems?

## Tuesday, December 23, 2014

### 351: Balance 6

Weights 1lb through 6lbs.
Where should we start?

## Monday, December 22, 2014

### 350: Primes

This little puzzle, via @mathmovesu, asks for three consecutive prime numbers.

Is the guess and check method the best way to go here?

Which prime numbers are candidates and which ones can we safely ignore?

## Sunday, December 21, 2014

### 349: Hole-in-One Insurance

If the average golfer is able to get a hole-in-one once in approximately 3000 rounds of golf (18 holes apiece), then what is the probability of any one of 100 average golfers getting a hole-in-one on the 5th hole during the weekend golf tournament?

What's the best way to find this out if you're the insurance company that will write this policy?

## Saturday, December 20, 2014

### 348: Homer's Pythagorean proposition

$1782^{12}+1841^{12}=1922^{12}$

Wait, didn't Fermat say this was impossible?
What's a three-second way to tell that this equation is false?

## Friday, December 19, 2014

### 347: Combinatorics

Consider eight objects. We will choose them one at a time, two at a time, three at a time, and so on.

Which of these will result in identical numbers of ways?
Why?

## Thursday, December 18, 2014

### 346: Casting the Play

The cast of a school play that requires 4 girls and 3 boys is to be selected from 7 eligible girls and 9 eligible boys.

• Will it be a different calculation if the boys are willing to play girls' parts, as in Shakespeare's time? If so, how will it be different?

## Wednesday, December 17, 2014

### 345: Fair or Foul?

Sullivan bought a die at the magic shop. He rolls it 155 times and gets the following results:
• ONE: twenty-eight times
• TWO: twenty times
• THREE: fifteen times
• FOUR: thirty-one times
• FIVE: thirty-two times
• SIX: twenty-nine times.
What is the probability he will get a 6 on the next roll?

## Tuesday, December 16, 2014

### 344: Monty Hall

Once upon a time, the world's smartest person (Marilyn vos Savant, IQ: 228) received a question for her newspaper column …
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? Craig. F. Whitaker, Columbia, MD

Marilyn's answer was surprising to many people. What do you think?

## Wednesday, December 10, 2014

### 343: Forces and Friction

Your teenage son has a fast car.

He knows that friction is determined by the weight of the car over the wheels, the "normal" force. He also knows that additional weight means that the car can't accelerate as fast, but he's also having problems with the rear tires spinning out.  He's convinced that having Fat Eddie sit in the back will help his quarter-mile time.

Will the extra weight help him or hurt him?

## Tuesday, December 9, 2014

### 342: Algebret

That takes care of four letters.
twenty-two to go ....

## Monday, December 8, 2014

### 341: Sensible or Not? Study Types

Give an example or a situation to bolster your position
1. The Department of Education conducted an observational study to determine the average salary of high school teachers in each of the United States
2. A paint company conducted a double-blind experiment to determine which of two types of exterior paint was more resistant to rain.
3. The lab conducted an experiment to determine whether the throat culture was positive for strep.
4. In a study of medications designed to slow the rate of balding in men, a placebo group had better results than the control group.
5. A meta-analysis was conducted to determine the population of New Zealand.
6. A case-control experiment was used to determine the average family size in Utah.

ch2.2

## Friday, November 21, 2014

### 318: Exponential Function

Is this enough information to find the equation?

## Thursday, November 20, 2014

### 317: Church

How many different math concepts can you use this church as an example for?

## Wednesday, November 19, 2014

### 316: Intersections

Which one is better?
Which one is easier to solve?

## Tuesday, November 18, 2014

### 315: Pizza Maker

Talk about the rate at which pizza sauce is being pumped out through the tube ...

## Monday, November 17, 2014

### 314: Stats Starter 2

Following on from yesterday's question, we have a puzzle from the same source.

Which of those central tendency statistics are necessary in order to find the missing numbers? (Necessary meaning that you can't find a particular number without it.) In order words, do we really need all five statistics?

Source.

## Sunday, November 16, 2014

### 313: Stats Starter 1c

Here is that same list of numbers.

You create a new problem this time ...
Can you create a solvable problem with just two hints?
Or do we need three?

Original source.

## Saturday, November 15, 2014

### 312: Stats Starter 1b

Here is a list of numbers.

You create the problem ...
What are some different statistics that you could give to a classmate (other than the ones below) yet still keep this a solvable problem, with the same answers as the original problem?

Original source.

Original problem:
mean = 76; range = 32; IQR = 21

## Friday, November 14, 2014

### 311: Stats Starter 1a

Here is a list of numbers.

What information would you NEED in order to determine the missing numbers?

Source.

If you want to solve the original problem yourself, you can go there and look for the rest of the problem and a discussion on finding the answers, but here is the set of numbers provided:
mean = 76; range = 32; IQR = 21

## Thursday, November 13, 2014

### 310: Missing Area

What things do we know?
What lines do we need to construct?
What unknowns do we need to plop a variable on?

Is this a problem best given to a Geometry class, an Algebra class, or Pre-Calculus class?

Source, For the Nguyen!

## Wednesday, November 12, 2014

### 309: Rational Cube Routes

Imagine a cube, 2 inches on a side ...

Actually, don't bother, I'll put one over there on the right side. ==>>

Now imagine if you were sitting on a vertex. How far is it (straight line distance) to the other vertices?

How many of those paths would be rational number distances?

What if you were on the midpoint of an edge and considering the paths to the vertices again. How many of those paths would have rational lengths distances?

And, no, I won't apologize for the pun. Pfft!

source:

## Tuesday, November 11, 2014

### 308: Congruent Lines?

Can two lines of infinite length be considered congruent?

## Monday, November 10, 2014

### 307: Expected Value vs Psychology

Which would you press ... and why?

## Sunday, November 9, 2014

### 306: Factorials and Perfect Squares

Let's examine the function g(n):

g(n) = smallest integer such that g(n)*n! is a perfect square.

How should we go about finding if there's a pattern in that?

## Saturday, November 8, 2014

### 305: Space and the Atmosphere

What kind of approach might give us the thickness of the atmosphere in the video?

## Friday, November 7, 2014

What's a good problem you can make based on this video?

## Thursday, November 6, 2014

### 303: Reworking Pythagoras

Pythagorean theorem:  a² + b² = c².

“In a right triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two legs.”

Here is the problem: does the figure whose areas we compare, drawn on the triangle's legs, have to be square?

Can there be other shapes – triangles, rhombuses, regular pentagons, etc. – that make the Pythagorean Theorem more generally true?

Source: Grant Wiggins, "The Problem of So-Called Problems - unpublished paper 2013"

## Wednesday, November 5, 2014

### 332: Which values of x do we choose? Polynomials

For the following functions, think about "How to graph like a math teacher." Math teachers want to sketch graphs quickly and efficiently and choose values of x that work "nicely" in the equation and  generate integer values of y, thus making it easier to graph.
• Which points are best found by inspection?
• Which points are best found by substitution?
• Which points are best found by symmetry?

$y = (x-2)^2(x+2)$

$y = (x-1)^3$

$y = x^3-8$

$y=x^3+3x^2$

$y=(x-3)^2(x+3)^2$

What are your favorite examples of this?

### 302: Systems of a Sort 8

Where should we start this shape substitution puzzle?

Source: Mimi! (I Hope This Old Train Breaks Down...)

## Tuesday, November 4, 2014

### 301: Systems of a Sort 7

What's the best way to proceed here?

## Monday, November 3, 2014

### 300: Systems of a Sort 6

Your teacher (me) is thinking about buying a new car. Presently, the cost of gas is $3.60 per gallon and he knows that he is going to commute to work each day and drive errands on the weekend ... about 300 miles per week. He wonders whether or not to buy a Corolla or a Prius. Currently, he drives a Ford Ranger pickup truck, which is paid off but is starting to have some expensive repair bills. If he averages those costs, it's about$200 per month.

Option 1: The Corolla costs $16,800 (about$400/month) and gets roughly 36 mpg.
Option 2: The Prius costs $24,200 (about$570/month) and gets roughly 49 mpg.
Option 3: The Ranger is paid off but has repairs (about \$200/month) and gets 21 mpg.
What should he do?

Prius:

Corolla:

## Sunday, November 2, 2014

### 299: Systems of a Sort 5

Is there an easy way to tell if those lines will have one solution, no solution, or an infinite number of solutions?

## Saturday, November 1, 2014

### 298: Systems of a Sort 4

This system of equations has a peculiar characteristic ... I think it is easier to solve it by analytical means than by using Desmos or a TI-84.

Do you agree?

What about it makes a graphical solution difficult?


text:
Although Thoreau goes to great lengths to get away from organized society, he is still unquestionably a humanist.

Show how Thoreau defends the very people and culture he spends much of his time avoiding.


text:
# Remote, brain region–specific control of choice behavior with ultrasonic waves

May 21, 2020

## Abstract

The ability to modulate neural activity in specific brain circuits remotely and systematically could revolutionize studies of brain function and treatments of brain disorders. Sound waves of high frequencies (ultrasound) have shown promise in this respect, combining the ability to modulate neuronal activity with sharp spatial focus. Here, we show that the approach can have potent effects on choice behavior. Brief, low-intensity ultrasound pulses delivered noninvasively into specific brain regions of macaque monkeys influenced their decisions regarding which target to choose. The effects were substantial, leading to around a 2:1 bias in choices compared to the default balanced proportion. The effect presence and polarity was controlled by the specific target region. These results represent a critical step towards the ability to influence choice behavior noninvasively, enabling systematic investigations and treatments of brain circuits underlying disorders of choice.

## INTRODUCTION

Noninvasive and reversible modulation of neuronal activity in specific brain circuits may allow us to diagnose and treat brain disorders in, targeted ways. Low-intensity ultrasound, applied to the brain noninvasively, can be used to modulate neural activity with spatial specificity superior to other noninvasive methods such as transcranial electrical or magnetic stimulation (15). The neuromodulatory potential of ultrasound has been highlighted in studies that targeted peri-motor regions of anesthetized rodents or rabbits. Brief, low-intensity stimuli lead to observable movements of the limbs or other body parts (613).

However, the enthusiasm about the neuromodulatory potential of ultrasound has recently been dampened by studies that called these effects into question (14, 15). In addition, such overt effects have not been observed in large mammals including humans. Only small changes in neural signals (1623) or small changes in reaction time or other metrics (2426) have been found. Yet, to make it truly useful, the approach would ideally provide neuromodulatory effects that are strong enough to manifest in behavior. For example, if clinicians are to determine which brain circuits drive a patient’s craving for an addictive drug, the neuromodulatory effects on a particular neural circuit should be potent enough to yield measurable changes in the subject’s choice behavior, i.e., whether the subject decides to use the drug or not.

Here, we engaged macaque monkeys in a choice task and report that brief pulses of low-intensity stimulation of specific brain regions have strong effects on the subjects’ choices. The ability to influence choice behavior noninvasively without using drugs could provide new ways to diagnose and treat disorders of choice.

## RESULTS

### Ultrasonic control of developing decisions

We engaged two macaque monkeys in a choice task that is often used in neurology to diagnose the impact of brain lesions such as those induced by stroke (2729). In this paradigm (Fig. 1A), one visual target is shown in the left and one is shown in the right visual hemifield, with a short, controlled delay between the onsets. Typically, in this task, healthy normal subjects tend to look at the target that appears first. Stroke or lesions of specific nodes of the oculomotor network, such as the frontal eye fields (FEF) or the lateral intraparietal area (LIP), strongly affect this behavior (2831). These oculomotor circuits preferentially represent targets in their contralateral visual hemifield (Fig. 1B). As a consequence, when neurons in these circuits are affected by a stroke, the contralateral visual hemifield is underrepresented, and subjects preferentially decide to look at the ipsilesional target (28, 29). These effects are also observed for other kinds of neural perturbations, such as when neuromodulatory agents are injected into these regions. For example, when muscimol—a potent neuroinhibitory drug—is injected into the left FEF of macaques (30), animals performing this task exhibit a strong ipsilateral (leftward) bias (Fig. 1C, red). In contrast, injecting a drug with opposite, disinhibitory effects such as bicuculline into the same region (30) leads to an opposite, contralateral bias (Fig. 1C, blue). Analogous results are obtained for neuromodulatory interventions into area LIP (3032). This task therefore provides a well-established framework that enables us to interpret the neuromodulatory effects of ultrasound or any other intervention for that matter. Excitatory interventions or stimuli bias subjects’ decisions in the contralateral direction, whereas inhibitory interventions bias subjects’ decisions in the opposite direction.

We used this framework to evaluate the polarity and size of the effects of ultrasound on neurons (Fig. 1A). In a given session, ultrasound was applied to the animals’ left or right FEF. Ultrasound was applied in blocks of three to six trials and was strictly interleaved with blocks of three to six trials in which ultrasound was not applied to mitigate potential effects of session time. The stimulus was applied for 300 ms starting at 100 ms before the onset of the first target to modulate neurons throughout the decision-making process. Given that animals responded on average within 171 and 263 ms (monkey A and B, respectively), this perturbation lasted long enough to influence a substantial portion of the process.

The ultrasound stimulus (Fig. 1D) had common neuromodulatory parameters (0.6 MPa, 270 kHz, 300 ms duration, 500 Hz pulse repetition frequency, and 50% duty cycle). We chose a relatively low carrier frequency of 270 kHz so that we could be confident that the stimulus would effectively penetrate the animals’ skull. As a consequence, the stimulus was relatively broad as indicated by our measurements of free field pressure (Fig. 1D). Full width at half maximum pressure was 10.5 mm in the lateral dimension and 21.8 mm in the axial direction below the skull. In monkey A (B), ultrasound was applied on average in 379 (481) trials [with a range of 103 to 909 (207 to 991)] trials per session; the stimulated trials constituted 50% of total trials.

Animals showed typical choice behavior in this task, being sensitive to the difference in target onset times (Fig. 2A, black curves). As in previous studies (2831), the earlier a target appeared before the subsequent target, the more likely the animal was to choose that target. Critically, transcranial ultrasound had a strong influence on this choice behavior (Fig. 2A, blue curves). In the trials in which ultrasound was applied to the left FEF (left column), animals were more likely to choose the rightward target. The effect reversed polarity when ultrasound was applied to the right FEF (right column): In this case, animals were more likely to choose the leftward target. These single-session effects were significant (P P=0.074). These examples suggest that ultrasound biases the animals’ decisions in the contralateral direction.

To quantify these effects across all sessions, we measured the proportion of rightward choices in ultrasound trials at the time point at which the animals chose both targets at equal proportion when not stimulated (see Materials and Methods). We did this separately for each session (dark bars in Fig. 2B) and for data pooled over sessions (light bars in Fig. 2B). We assessed significance using a two-tailed t test and a randomization test (see Materials and Methods), respectively. This analysis shows that targeting the left FEF increased the proportion of rightward choices (Fig. 2B, left). The effect was especially strong in monkey A, who chose the rightward target, when stimulated, in 65.9% of cases at the point of otherwise equal preference (50%) when averaged over the individual sessions, and in 67.0% cases when the data of the eight sessions were pooled together. This differed significantly from 50% across the eight sessions in this animal (P=0.0015, t7=5.0, two-tailed t test), as well as when the data were pooled together (P P=0.012, t7=3.4 and P P=0.0002, t15=4.9). In four sessions, the effect exceeded 75% (3:1 choice bias). As in the single-session examples (Fig. 2A, right column), the effect reversed polarity in the trials in which the right FEF was stimulated (Fig. 2B, right). Monkey A chose the rightward target in 43.1% (session average) or 45.4% (pooled) of cases. The effect was not significant when assessed over the eight sessions but was significant when the data were pooled together (P=0.0072, randomization test). Monkey B chose the rightward target in 41.3% (42.7%) of cases, and this effect was significant (P=0.0024, t7=−4.6 and P P=0.00092, t15=−4.1). These effects are graphically summarized at the bottom of Fig. 2B. Stimulation of the left FEF significantly increased the proportion of rightward choices, whereas stimulation of the right FEF significantly increased the proportion of leftward choices. The contralateral nature of these shifts suggests that ultrasound positively biased the decision process represented by neurons within the stimulated region.

There was a difference in the way the two monkeys were rewarded after completing a trial. Monkey A was rewarded for choosing either target, not only for choosing the target that appeared first. This enabled the animal to choose without the guidance of target timing. A consequence was that the animal exhibited a directional bias (Fig. 2A, top row, black curves). Because such a bias is common in this task (33), to exclude its potential influence on our results, monkey B was only rewarded for choosing the first target (see Materials and Methods). This modification (31) greatly reduced this bias. Nevertheless, Fig. 2B shows that, in retrospect, this additional control was not strictly necessary—the effects point in the same direction in both animals and are on average of comparable size.

### Nature of the effect

We next investigated how rapidly the effect emerges and whether it is cumulative or, in contrast, whether there is an adaptation. Our task interleaves blocks of stimulated and nonstimulated trials (each three to six trials in duration). This block design enabled us to assess the dynamics of the ultrasonic effects as a function of the number of successively stimulated trials. To do so, we pooled data across right and left FEF stimulation sites and present the average proportion of contralateral choices as a function of trial number within a stimulated and nonstimulated block (Fig. 2A). The figure reveals that the biasing effect of ultrasound emerges immediately, in the first stimulated trial within a stimulation block, and gradually diminishes in amplitude. The effect is significant (two-tailed two-sample proportion test) in the first (P P P=0.010), and fourth (P=0.039) stimulated trials within a block [nonsignificant (P> 0.16) thereafter].

The analysis of Fig. 3A reveals that the choice behavior–controlling effects of ultrasound are stronger for the initial first stimuli within a stimulation block, before an adaptation occurs. When we repeated the analysis of Fig. 2B using the first three trials within each stimulation block, the biasing effects of ultrasound are stronger (Fig. 3B). For instance, the effect was larger than 80% in 6 of the 16 sessions of left FEF stimulation. In these sessions, the stimulus biased the animals’ choices in a ratio greater than 4:1. In all subsequent analyses, we use data of all trials and not only the initial stimulation trials in which the effect was strongest.

We asked whether ultrasound only biased choice or whether it, in addition, influenced perception. In particular, we asked whether ultrasound shifted the decision curves along the horizontal axis—which would represent a bias in choice, or changed the curves’ slope—which would represent an impaired ability to distinguish the two targets. To do so, we fitted the decision curves with a two-parameter sigmoid fit (31), separately for the stimulated and nonstimulated decision curves within each session. We found that ultrasound shifted the curves along the horizontal axis (Fig. 4). Left FEF stimulation shifted the decision curves on average by −7.6 ms (Fig. 4A, left), and this effect was highly significant across the sessions (P t15=−4.3; two-sided t test). A leftward shift indicates, for a given difference in target onset times, that the animals were more likely to choose the rightward target when stimulated. The effect reversed polarity during right FEF stimulation, shifting the decision curves by 5.6 ms across the sessions. Moreover, this effect was highly significant (P=0.0064, t15=3.2). A rightward shift indicates, for a given difference in target onset times, that the animals were more likely to choose the leftward target when stimulated. These effects were significant within individual sessions (Fig. 4B). Stimulation of the left FEF produced a significant horizontal shift of decision curves in 7 of 16 sessions, by an average of −13.4 ms (−7.6 ms across all 16 sessions). Stimulation of the right FEF produced a significant horizontal shift in 3 of 16 sessions, by an average of 11.1 ms ( 5.6 ms across all 16 sessions). These shifts corroborate the finding of Fig. 2 that ultrasound stimulation biased the animals’ choices in the contralateral direction. Figures S1 and S2 show effects for each individual session.

Compared to the biasing effects, ultrasound had only a mild effect on the slope of the curves. Significant shallowing was observed only during left FEF stimulation (a mean change of −0.0095; P=0.032, t15=−2.4) and not during right FEF stimulation (P=0.22, t15=−1.3). The effect was significant in only 1 of 16 sessions (Fig. 4B). The lack of substantial shallowing suggests that ultrasound did not notably impair the animals’ ability to distinguish the onsets of the two targets.

### The effect is region specific

A concern raised by previous studies (14, 15) is that the stimulus could be generating a radiation force whose 500-Hz modulation is audible to animals and that this auditory perception could underlie the observed effects. This is unlikely as the differential effects of right and left FEF stimulation (Figs. 2 and 4) demonstrate that the effects are specific to the stimulated regions. However, there could be a directional effect of auditory attention. To provide an additional control for such indirect confounds, we applied the same stimuli to control regions 10 mm more posterior to FEF, i.e., the motor cortex. Motor cortex is not involved in perceptual choice behavior, and so, in this case, there should be no effects on animals’ choices. We collected data in 11 sessions of left motor cortex stimulation and 11 sessions of right motor cortex stimulation. Besides the change in the stimulation location, data were collected in the same way as with the FEF stimulation. In contrast to FEF stimulation, motor cortex stimulation did not elicit significant biases in choice behavior (Fig. 5). There were no significant shifts from equal preference (dark bars; left motor cortex: P=0.43, t10=−0.82; right motor cortex: P=0.36, t10=0.95). This was also the case when data over the individual sessions were pooled together (light bars; P> 0.67 for either monkey and hemisphere, randomization test). We further increased the power of this analysis by pooling data across left and right motor cortex stimulation sessions, thus obtaining n=22 sessions. There was no significant shift from equal preference (P=0.63, t21=0.49). We also tested for a potential reversal of effect polarity upon reversal of the stimulated hemisphere—just like we observed in FEF—by pooling positively signed left motor cortex shifts with negatively signed right motor cortex shifts. This analysis also did not yield a significant effect (P=0.23, t21=−1.24). In addition, stimulation of left and right motor cortices did not significantly change the horizontal position or slope of the decision curves within single sessions.

## DISCUSSION

There has been considerable excitement about low-intensity ultrasound as a neuromodulatory approach because it is currently the only modality that can noninvasively target compact neural circuits, including deep nuclei, in the human brain. However, this enthusiasm has recently been curbed by studies reporting weak effects and two studies questioning the very validity of these effects (14, 15). Using nonanesthetized primates performing a choice task, we found that transcranial ultrasound can have substantial and region-specific effects on target circuits. Brief pulses of low-intensity ultrasound strongly, controllably, and reversibly influenced the subjects’ choice behavior.

The effect was sizable, leading to about 2:1 bias in choice behavior (Fig. 2). The first initial stimuli within a stimulation block had particularly strong leverage on the choice behavior (Fig. 3), leading to up to an about 4:1 bias. The effect was attained using stimuli with specifications that fall within the U.S. Food and Drug Administration (FDA) guidelines on diagnostic ultrasound (see Materials and Methods) (34, 35). No adverse short- or long-term effects were observed.

Stimulation of left/right FEF shifted the animals’ choices rightward/leftward (Figs. 2, 3, and 4). These data suggest that our stimulus positively biased the decision process represented by neurons within the stimulated region. According to the drift-diffusion model that is often used to describe behavior in two-alternative choice tasks (see Fig. 6 for a rendering of the concept), an increase in the choice of an option could be due to three factors. First, ultrasound may bias the starting balanced state (Fig. 6, panel 1), even before a stimulus is presented. Second, ultrasound may influence the speed of the drift toward a choice (represented by a particular weight in panel 2). Last, ultrasound may change the threshold (represented by squares of a particular height in panel 3) at which a decision is concluded. Future studies should deliver ultrasound specifically at these individual stages of the decision process to illuminate the exact nature of the biasing effect.

The primate model, which has a sizable cranium, enabled us to investigate the local specificity of the effect. We found that mirroring the stimulation site (left versus right FEF) reversed the effect polarity. This constitutes a double dissociation of the effect through different brain regions. In addition, there were no effects when stimulating control regions that are not involved in oculomotor choice, i.e., right and left motor cortices (Fig. 5). These data provide a negative control for potential generic artifacts that can be associated with ultrasound (14, 15).

Our experimental design using blocks of sonicated trials interleaved with nonsonicated trials allowed us to assess the effect progression as a function of the number of consecutive stimuli (Fig. 3A and fig. S3). A neuromodulation effect can be constant in time, cumulative (increasing with each additional intervention), or diminishing (decreasing with each additional intervention). We found evidence for the latter kind. The effect emerged immediately, within the first stimulated trial. It then gradually decreased in size until becoming nonsignificant at the fifth consecutive stimulus. This indicates that the system adapted to the repetitive stimulation. The nature of this adaptation is currently unknown. One possibility is that the adaptation occurs at the molecular level, whereby the molecular machinery gradually loses sensitivity to repetitive excitation. This possibility is likely given the emerging view that the effects of ultrasound on neurons are of a mechanical kind. In particular, the mechanical forces associated with propagating ultrasound displace membranes and, in this way, open mechanosensitive ion channels (3638). It has been demonstrated that mechanosensing molecular machinery adapts to repeated mechanical stimulation (39).

This study provides several recommendations for future applications of ultrasonic neuromodulation in animals and humans. First, the effects of ultrasonic neuromodulation on behavior are likely to be maximized when a subject is making a decision. The delicate balance of weighing decision options renders neural circuitry sensitive to perturbation (Fig. 6). This has been demonstrated in previous studies (30, 31) and in the present study (Fig. 2A and figs. S1 and S2), in which the stimulation effects are particularly prominent near the point of equal preference. Second, anesthesia should be avoided. Anesthesia dampens neuronal excitability and susceptibility to stimulation. Both the anesthesia level (41) and the kind of anesthesia (42) affect ultrasound-mediated responses, and some authors claim responses only under specific anesthesia conditions (8, 11). There may even be little or no neuromodulatory effects under anesthesia (14, 15). Third, the ultrasonic stimulus should be relatively long (several dozens of milliseconds) and applied before the respective neuronal pools are engaged in a given task. This is because the effects of ultrasound take time to emerge (8, 37). This effect may be due to an integration of membrane voltage toward a threshold for activity (43). Fourth, stimulation should be applied relatively infrequently. This reflects our finding that repetitive stimulation diminishes the neuromodulatory effects (Fig. 3A and fig. S3). The effects of ultrasound on neurons are, in part, of mechanical nature (37), and it is known that frequent mechanical stimuli lead to an adaptation at the level of cells and molecules (39). Fifth, good acoustic coupling and real-time monitoring is paramount. Acoustic coupling from a transducer to skin has been, in most neuromodulation studies, achieved using a coupling cone. In our case, the cone was positioned over the temporal muscle. The muscle is strongly developed in male macaques and is engaged each time an animal receives juice reward. This movement could have led to a loss of coupling or at least to variability in the targeted FEF subregions. Because specific FEF subregions code for specific target locations (44), variability in targeted FEF subregions could have led to variable effects on decision curves. Some sessions showed strong effects and some sessions showed no effect, and decision curves of single sessions were affected in various ways (figs. S1 and S2). These methodological issues can be mitigated by using a system that (i) couples ultrasound to the skin using water and (ii) operates under real-time magnetic resonance imaging (MRI) guidance (45).

The finding that monkey A, who was rewarded for looking at either target, showed effects of the same polarity as monkey B suggests that ultrasound could be used to influence also internally motivated choices. This could turn ultrasound into a powerful tool to study the neural circuits involved in normal and aberrant choice behaviors. In particular, disorders of choice that include addictive, binge eating, and compulsive behaviors commonly involve deep brain regions. Selective targeting of such regions is a major strength of ultrasound (45). For instance, the nucleus accumbens shell has been causally linked to the decision whether to approach food in rodents (46). Ultrasound could be used to test whether this nucleus controls food intake decisions in humans. Moreover, modulation of nuclei and pathways in limbic circuits and the basal ganglia could lead to new targets for psychiatric conditions that manifest in impulsive or compulsive choices. The hardware to deliver ultrasound into the deep brain of humans is readily available and operational, attaining about a 4-mm full width at half maximum in the deep brain when using hemispherical arrays consisting of hundreds of individually controllable transducer elements (45). There are, therefore, tantalizing opportunities to apply ultrasonic neuromodulation to noninvasively modulate choice behavior in humans, with first applications aimed at determining the set of circuits involved in a given disorder in a given individual.

In summary, we investigated whether ultrasound, a pressure wave applied remotely, can modulate brain activity to an extent of producing notable effects on forming decisions. We found that brief, low-intensity ultrasonic pulses remotely delivered into oculomotor circuits of nonhuman primates markedly influence perceptual decisions. The presence and polarity of the effect were controllable by a specific target region. This result takes us a step closer to being able to modulate, noninvasively and reversibly, neuronal activity in specific brain circuits. This could open the way to future systematic studies of brain function in humans and to targeted personalized treatments of brain disorders.

## MATERIALS AND METHODS

### Experimental design

To complete a trial, monkeys were allowed to look at either target. In the first month of training, monkeys were rewarded to look at the target that appeared first. Following this initial training and during data collection, monkey A was rewarded for looking at either target. This was to allow the animal to choose without the guidance of target timing. This task led to strong directional biases over individual sessions, as reported previously (33). If a session resulted in a strong directional bias, such that the decision curve did not reach full saturation within the used range of stimulus onset delays, we adjusted the range for the next session such that the point of equal preference lay in the middle of the range. This adjustment was only necessary for monkey A and was applied five times over 22 sessions. We were concerned that these biases may unpredictably interact with the stimulation effects. Therefore, in monkey B, reward was delivered only for choosing the target that appeared first. This helped to mitigate directional biases (fig. S2).

### Ultrasonic stimulation

Ultrasound was applied in blocks of three to six trials (the specific number was drawn from uniform distribution bounded between three and six) and was strictly interleaved with blocks of three to six trials in which ultrasound was not applied. We stimulated the macaque FEF with a single-element transducer (H-115, diameter 64 mm, Sonic Concepts). The transducer was geometrically focused to 63 mm using a coupling cone filled with agar. The height of the cone was chosen such that the geometric focus was located 5 mm below the skull, to ensure that ultrasound stimulated neurons within the entire depth of the arcuate sulcus. The stimuli (270 kHz, 300 ms duration, 500 Hz pulse repetition frequency, 50% duty cycle; Fig. 1D) were generated using a commercial function generator (33520B, Keysight) and subsequently amplified using a commercial amplifier (A150, E&I). The output pressure maximum was set to 0.6 MPa. The pressure field (Fig. 1D) was characterized in vitro in free field, using AIMS III (Onda) water tank filled with distilled and degassed water. The same coupling cone filled with agar gel as that used in the main experiment was used in these measurements; no ex vivo skull was present. The measurements were taken using a calibrated fiber-optic hydrophone (Precision Acoustics). The distribution of the pressure field was measured using a robotized moving stage (AIMS III) and characterized in 1-mm (2-mm) steps in the lateral (axial) dimensions (Fig. 1D). The skull may act as an acoustic lens and so the focal area attained in the experiments may be tighter. For this stimulus, the spatial-peak pulse-average intensity in water (free field) is ISPPA=11.6 W/cm2. This value is well below the FDA track 3 recommendation of ISPPA=190 W/cm2 (34, 35). The spatial-peak time-average intensity, ISPTA=581 mW/cm2, is also below the FDA recommendation of ISPTA=720 mW/cm2. The attenuation of the skull further lowers these values. A previous study (24) measured a pressure attenuation of 42 ± 8% through one skull; another study (19) simulated a pressure attenuation of 46 and 23% in two monkeys. On the basis of these data, the estimated peak pressure at focus below the skull of our animals is between 0.32 and 0.46 MPa.

During the experiment, the animals’ hair was shaved, and degassed ultrasound gel was applied on the skin to mediate acoustic coupling between the agar-filled coupling cone and the skin. The FEF target was localized using anatomical MRI images.

Each session was conducted on a separate day; left and right stimulation sessions were also always performed on separate days.

### Characterization of decision curves

We fitted the decision curves obtained in each session with a sigmoid function. The fit was performed separately for stimulated and nonstimulated data (e.g., blue and black curves in Fig. 2A). We used the same four-parameter fit as a previous study (31). This fit is mathematically equivalent to logistic fit, with the exception that it features two additional parameters to also capture vertical properties (vertical scale and vertical position) of the decision curves

$P(x)=11e−slope(x−horizpos)vertscalevertposFig. 4, “horizontal shift” is the difference in the fitted horizpos values of the stimulated and nonstimulated curves.$

From this equation, the point of equal preference used in the analysis of Figs. 2B, 3B, and 5, x50, is determined as

P(x50) across the individual sessions. In Figs. 2B and 3B, this analysis was performed for each individual session (dark bars) and for choice data pooled over the sessions (light bars). In the latter case, to assess significance, we used a randomization test, described below.

### Statistical analysis

To assess how the fitted parameters changed between stimulated and nonstimulated trials in each session (Fig. 4B), we performed a randomization test. In this test, the nonstimulated binary choice data for each difference in target onset times were sampled, with replacement, 10,000 times. Each of these resampled decision curves were fitted with a sigmoid function. The fitting procedure was the same as above with the exception that we only used horizpos and slope as parameters. The main conclusions remain the same regardless of whether we use two or four parameters, but using two parameters helped to increase the statistical power of the analysis. The fits produced a null distribution of 10,000 values for each parameter. We then fitted the two parameters to the stimulated curve and evaluated the probability that the parameters were drawn from the respective null distributions. If the probability was less than 0.01 for a given parameter, Bonferroni-corrected for the number of sessions, the change was considered significant.

We also performed a randomization test when assessing the significance of effects for data pooled together over the sessions. First, we determined the average point of equal preference (x50) for the nonstimulated decision curve. We then sampled the binary choice data of the stimulated curve for each difference in target onset times 10,000 times, with replacement. We computed P(x50) for each of these 10,000 resampled decision curves. The test returns the probability that 0.5, i.e., equal choice proportion, was drawn from the P(x50) distribution.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

1. FDA, Marketing clearance of diagnostic ultrasound systems and transducers. Guidance for Industry and Food and Drug Administration Staff (2019).

Acknowledgments: We thank S. Fong and S. Hyde for technical and procedural assistance. Funding: This work was supported by the NIH grant K99NS100986 (J.K.), Stanford Medicine Dean’s Fellowship (J.K.), and Howard Hughes Medical Institute (J.B., T.M., and W.N.). Author contributions: J.K. designed the task, performed the experiments, analyzed the data, and wrote the paper. J.B. designed the task and edited the paper. P.Y. performed the experiments. K.B.P. designed the task and edited the paper. T.M. designed the task and edited the paper. W.N. designed the task and edited the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data such as saccadic traces may be requested from the authors.


text:
# How do I find the extraneous solution of sqrt(x-3)-sqrt(x)=3?

Sep 28, 2014

$\sqrt{x - 3} - \sqrt{x} = 3$

by subtracting $\sqrt{x - 3}$,

$R i g h t a r r o w - \sqrt{x} = 3 - \sqrt{x - 3}$

by squaring both sides,

$R i g h t a r r o w x = 9 - 6 \sqrt{x - 3} + x - 3$

by cancelling out $x$'s,

$R i g h t a r r o w 0 = 6 - 6 \sqrt{x - 3}$

by dividing by 6,

$R i g h t a r r o w 0 = 1 - \sqrt{x - 3}$

by adding $\sqrt{x - 3}$,

$R i g h t a r r o w \sqrt{x - 3} = 1$

by squaring both sides,

$R i g h t a r r o w x - 3 = 1$

$R i g h t a r r o w x = 4$
Even though all implications above are valid, some of them are not reversible. As a result, $x = 4$ we found does not satisfy the original equation, which makes it extraneous.


text:
# Triangle Inequality/Examples/4 Points

## Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $x, y, z, t \in A$.

Then:

$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$

## Proof

We have that $\map d {x, z}$, $\map d {y, t}$, $\map d {x, y}$, $\map d {z, t}$ are themselves all real numbers.

Hence the Euclidean metric on the real number line can be applied:

 $\ds \size {\map d {x, y} - \map d {z, t} }$ $\le$ $\ds \size {\map d {x, y} - \map d {y, z} } + \size {\map d {y, z} - \map d {z, t} }$ $\ds$ $\le$ $\ds \map d {x, z} + \map d {y, t}$ Reverse Triangle Inequality twice

$\blacksquare$


text:
# Mean

Mean or average or arithmetic mean is one of the representative values of data. We can find the mean of observations by dividing the sum of all the observations by the total number of observations.

Mean of raw data:

If x1, x2, x3, ……. xn are n observations, then

Arithmetic Mean = (x1, x2, x3, ……. xn)/n

= (∑xi)/n

∑ (Sigma) is a Greek letter showing summation

1. Weights of 6 boys in a group are 63, 57, 39, 41, 45, 45. Find the mean weight.

Solution:

Number of observations = 6

Sum of all the observations = 63 + 57 + 39 + 41 + 45 + 45 = 290

Therefore, arithmetic mean = 290/6 = 48.3

Mean of tabulated data:

If x1, x2, x3, x4, ……. xn are n observations, and f1, f2, f3, f4, ……. fn represent frequency of n observations.

Then mean of the tabulated data is given by

= (f1 x1 + f2 x2 + f3 x3 + ……. fn xn)/(f1 + f2 + f3 + …… fn) = ∑(fixi)/∑fi

2. A die is thrown 20 times and the following scores were recorded 6, 3, 2, 4, 5, 5, 6, 1, 3, 3, 5, 6, 6, 1, 3, 3, 5, 6, 6, 2.

Prepare the frequency table of scores on the upper face of the die and find the mean score.

Solution:

 Number on the upper face of die Number of times it occurs (frequency) fixi 1 2 1 × 2 = 2 2 2 2 × 2 = 4 3 5 3 × 5 = 15 4 1 4 × 1 = 4 5 4 5 × 4 = 20 6 6 6 × 6 = 36

Therefore, mean of the data   = ∑(fixi)/∑fi

= (2 + 4 + 15 + 4 + 20 + 36)/20

= 81/20

= 4.05

3. If the mean of the following distribution is 9, find the value of p.

 X 4 6 p + 7 10 15 f 5 10 10 7 8

Solution:

Calculation of mean

 xi fi xifi 4 5 20 6 10 60 p + 7 10 10(p + 7) 10 7 70 15 8 120

∑fi = 5 + 10 + 10 + 7 + 8 = 40

∑ fixi = 270 + 10(p + 7)

Mean = ∑(fixi)/∑fi

9 = {270 + 10(p + 7)}/40

⇒ 270 + 10p + 70 = 9 × 40

⇒ 340 +10p = 360

⇒ 10p = 360 - 340

⇒ 10p = 20

⇒ p = 20/10

⇒ p = 2

Mean of grouped data:

While calculating the mean of the grouped data, the values x1, x2, x3, ……. xn are taken as the mid-values or the class marks of various class intervals. If the frequency distribution is inclusive, then it should be first converted to exclusive distribution.

4. The following table shows the number of plants in 20 houses in a group

 Number of Plants 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 12 - 14 Number of Houses 1 2 2 4 6 2 3

Find the mean number of plans per house

Solution:

We have

 Number of Plant Number of Houses (fi) Class Mark (xi) fixi 0 - 2 1 1 1 × 1 = 1 2 - 4 2 3 2 × 3 = 6 4 - 6 2 5 2 × 5 = 10 6 - 8 4 7 4 × 7 = 28 8 - 10 6 9 6 × 9 = 54 10 - 12 2 11 2 × 11 = 22 12 -14 3 13 3 × 13 = 39

∑fi = 1 + 2 + 2 + 4 + 6 + 2 + 3 = 20

∑fi xi =1 + 6 + 10 + 28 + 54 + 22 + 39 = 160

Therefore, mean = ∑(fixi)/ ∑fi = 160/20 = 8 plants

Statistics


text:
# zbMATH — the first resource for mathematics

Spanning trees with many leaves. (English) Zbl 0986.05030
In this paper, it is shown that for every $$t\geq 2$$, and every graph $$G=(V,E)$$ with $$|V|\neq t+2$$ and $$|E|\geq |V|+ t(t-1)/2$$, $$G$$ has a spanning tree with more than $$t$$ leaves. This bound is sharp. Similar results are given for the case that $$|V|=t+2$$.

##### MSC:
 05C05 Trees
##### Keywords:
spanning tree; leaves
Full Text:
##### References:
 [1] Griggs, Discrete Math 104 pp 167– (1992) · Zbl 0776.05031 [2] and Spanning trees with many leaves, lecture at SIAM Discrete Math, Conf., San Francisco, 1988. [3] The maximum leaf spanning tree problem for cubic graphs is NP-complete, Institute for Mathematics and its Application, preprint, 1998. [4] Storer, Inform Process Lett 13 pp 8– (1981) · Zbl 0482.05031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.