John Von Neumann (1903-1957)
Hans Bethe:
Academic seminars
(10 levels) who can understand
(This is sexism, my apology.)
Level 1: my mother
Level 2: my wife
Level 7: myself
Level 8: John and the Speaker
Level 9: John, (the Speaker didn't)
http://www.scidiv.bcc.ctc.edu/Math/vonNeumann.html
The man who knew 28%
of mathematics.
Level 10: not even Johnny
Random Numbers Generators
-- John Von Neumann --
“Any one who considers arithmetical method of producing
Random numbers is, of course, in a state of sin…..”
“… there is no such thing as a random number – there are only
methods to produce random numbers, and an arithmetical
procedure is of course not such a method…”
“..... a problem we suspect of being solvable by
random methods may be solvable by some rigorously
defined sequence….”
Two dice
How to use them to generate random numbers?
Roll one die: 4, 3, 2, 2, 6, 3, 5, 4, 6,...
Roll two dice: 4, 8, 5, 9, 10, 2, 8,...
Are they really random?
But, there is no die in any computer.
Ludwig Wittgenstein
(1889-1951)
“Turing Machines are
human that compute.”
“In logic nothing is accidental”
Image from http://www.ags.uci.edu/~bcarver/wgallery.html
Using Computers
Pseudo-Random numbers.
A new random number will be generated based
on some old numbers.
The 1st one is based on a seed.
X0 X1 X2 X3
Criterion
Xi-1 Xi Xi+1
1. How long is the period?
2. Is that sequence sufficiently random ?
Shift digits Method
X0 X1 X2 X3
Xi-1 Xi Xi+1
Suppose we want to have a sequence of
random numbers between 0 and 99999
12345 12345  12345 = 152399025
23990  23990 = 575520100
55201  55201 = 3047150401
Xi+1 =(Xi  Xi / 100 ) mod 100,000
X mod m = the remainder of X divided by m
e.g. 9 mod 5 = 4, 5 mod 2 = 1, 9 mod 3 = 0.
Linear Congruential Method
Let a, c, and m be integer.
a : multiplier
c: increment
m: modulus
Xi+1 =(aXi + c) mod m
X mod m = the remainder of X divided by m
e.g. 9 mod 5 = 4, 5 mod 2 = 1, 9 mod 3 = 0.
Theorem:
Xi+1 =(aXi + c) mod m
The linear congruential sequence has a period of length m iff
1. c is relatively prime to m;
2. (a - 1) is a multiple of every prime p dividing m;
3. (a - 1) is a multiple of 4, if m is a multiple of 4.
The first proof was due to M. Greenberger in 1961 for m = 2n ;
The general case was proven by Hull and Dobell in 1962.