congruential

A logic satisfying this weaker replacement property is called selfextensional by Wójcicki and congruential by Humberston (Humberstone, 2005).

Propositional Consequence Relations and Algebraic Logic Jansana, Ramon 2006

“We subtracted two sets of big numbers from our gigantic congruential multiplier*, came out with a positive number, therefore we can claim it is unprecedented and predict the behaviour of a coupled non-linear system a hundred years from now in spite of having no actual hard data”.

"We cannot make claims as to the 1990s being the warmest decade." « Climate Audit 2005

Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators (MCG) How to choose m, a?

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The linear congruential generator associated with the triplet (m, a, c) is given by: S = {0, ..., m − 1} for c >

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators Demonstration (2/3) 1 If α0 + · · · + αd − 1 ad − 1 ≡ 0 mod m, σ = α0 un +

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators Bad high dimension properties d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 d = 9 d =

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators Problem with high dimension Suppose that one want to sample uniformly the hypercube [0, 1] d.

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Linear congruential generators Properties It is the fi rst class of PRNGs for which a mathematical theory has been built, They are fast, They are easy to implement, They are (still) widespreads (rand (), drand48 () ...)

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators Bad high dimension properties: Marsaglia's theorem, from [3] Let d

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Pseudo-random number generators Uniform PRNGs Non-uniform PRNGs Multiplicative congruential generators Demonstration (1/3) To demonstrate Marsaglia's theorem, we need the following result from number theory: Theorem [4, Theorem 449] Let ξ0, ..., ξd − 1 be d linear forms d − 1 ξi = j = 0 ki, j ti with real coe ffi cients, such that: k0,0 ··· k0, d − 1 ∆ = ....

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