John Von Neumann 19031957

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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)
Level 10 not even Johnny
http//www.scidiv.bcc.ctc.edu/Math/vonNeumann.html
The man who knew 28 of mathematics.
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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.
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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.
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Ludwig Wittgenstein (1889-1951)

• Turing Machines are human that compute.

In logic nothing is accidental
Image from http//www.ags.uci.edu/bcarver/wgaller
y.html
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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
X3
X2
Xi1
Xi
Xi-1
Criterion
1. How long is the period?
2. Is that sequence sufficiently random ?
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Shift digits Method
X0
X1
X3
X2
Xi1
Xi
Xi-1
Suppose we want to have a sequence of random
numbers between 0 and 99999
12345 ? 12345 152399025
23990 ? 23990 575520100
12345
55201 ? 55201 3047150401
Xi1 (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.
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Linear Congruential Method
Let a, c, and m be integer.
a multiplier c increment m modulus
Xi1 (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.
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Theorem
Xi1 (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.