by gill1109 » Sun Jan 10, 2021 10:14 pm
FrediFizzx wrote:Last time you said I was wrong it was YOU that was wrong. Are ya sure you want to go that route again and look more foolish?
Well, I do make mistakes from time to time, and when I learn about them, I admit them and try to repair any damage found. Close to 70 years old, I make more mistakes now, and I'm slower to "get" things, than when I was 30.
But no matter. This is your model:
Take two large wooden disks and colour half of each black, half white. Fix to a wall. There's a pointer painted on the disks, near the edge, in the middle of the black half of the circumference. Painted on the wall, just above the top of each disk, is another pointer. Around the side of the disk, painted on the wall, equally spaced, are the numbers 1 to 360; 360 on top. Spin each disk. Wait till it stops. The pointers painted *in* the disks each point to a number alpha, beta between 1 and 360, painted on the wall. The pointers at the top of the disk painted *on the wall* point either to black, or to white, *in *the disk. That defines A = +/- 1 and B = +/- 1. You repeat this fairground game many, many times, and average A times B for each value of delta = alpha - beta.
That's the curve you drew.
Exercise: compute it analytically. Probably Mathematica can do it by computer algebra if you are clever with Mathematica! Good luck.
Hint: do the continuous case, and simplify the double integration over alpha and beta by a change of variables to alpha and delta.
Image: two fairground "wheels of fortune" which have come to rest at A = +1 (black on top), B = -1 (white on top), alpha = 330 (approx), beta = 220 (approx)
[quote="FrediFizzx"]Last time you said I was wrong it was YOU that was wrong. Are ya sure you want to go that route again and look more foolish?
[/quote]
Well, I do make mistakes from time to time, and when I learn about them, I admit them and try to repair any damage found. Close to 70 years old, I make more mistakes now, and I'm slower to "get" things, than when I was 30.
But no matter. This is your model:
Take two large wooden disks and colour half of each black, half white. Fix to a wall. There's a pointer painted on the disks, near the edge, in the middle of the black half of the circumference. Painted on the wall, just above the top of each disk, is another pointer. Around the side of the disk, painted on the wall, equally spaced, are the numbers 1 to 360; 360 on top. Spin each disk. Wait till it stops. The pointers painted *in* the disks each point to a number alpha, beta between 1 and 360, painted on the wall. The pointers at the top of the disk painted *on the wall* point either to black, or to white, *in *the disk. That defines A = +/- 1 and B = +/- 1. You repeat this fairground game many, many times, and average A times B for each value of delta = alpha - beta.
That's the curve you drew.
Exercise: compute it analytically. Probably Mathematica can do it by computer algebra if you are clever with Mathematica! Good luck.
Hint: do the continuous case, and simplify the double integration over alpha and beta by a change of variables to alpha and delta.
[img]https://gill1109.files.wordpress.com/2021/01/twodisks.jpg[/img]
Image: two fairground "wheels of fortune" which have come to rest at A = +1 (black on top), B = -1 (white on top), alpha = 330 (approx), beta = 220 (approx)