By Arkhipov B.

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**Additional info for Numerical Modeling of Pollutant Dispersion and Oil Spreading by the Stochastic Discrete Particles Method**

**Sample text**

A branch of a tableau is closed if it contains an explicit contradiction: b o t h T k Z and F k Z, for some k and Z. If every branch is closed, the tableau itself is closed. Intuitively, when we begin a tableau with F 1 X we are supposing there is some possible world, designated by 1, at which X fails to hold. A closed tableau represents an impossible situation. So the intuitive understanding of a tableau proof is t h a t X cannot fail to hold at any w o r l d - - X must be valid. P r o p e r soundness and completeness proofs can be based on this intuition but there is not space here to present them.

Bertrand Russell, Herbrand's Theorem, and the Assignment Statement 17 First Possibility The formula OP(c) asserts that, whatever c means, it has the " 0 P " property. A reasonable way of formalizing this in models is to allow the occurrence of free variables, together with a valuation function to assign values to them. Thus we broaden the machinery a bit, and write A/l, F tf-v X to mean: X (which may contain free variables) is true at world F of model A/I, with respect to valuation v which assigns values to the free variables of X.

A valuation in A/t is a mapping v assigning to each variable x some member v(x) in the domain of the frame underlying the model. Note that valuations are not world dependent. We say a term t designates at F if t is a variable, or if t is a constant symbol and Z(t, F ) is defined. If t designates at F we use the following notation: Bertrand Russell, Herbrand's Theorem, and the Assignment Statement 19 (v 9 2:)(t, F) = ~ v(x) if t is the variable x 27(c, F) if t is the constant symbol c Finally we must define f14, F iFv ~: formula ~ is true at world F of model Ad with respect to valuation v.

### Numerical Modeling of Pollutant Dispersion and Oil Spreading by the Stochastic Discrete Particles Method by Arkhipov B.

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