Consider the initial-value problem for a constant-coefficient symmetric hyperbolic system with initial-values vanishing in a half-space. Consider also a finite difference operator consistent with the system. Conditions are given in terms of the orders of dissipation and accuracy which ensure that the solution of the discrete problem tends to zero exponentially with the mesh-width in half-spaces...

A fundamental property of convex functions in continuous space is that the convexity is preserved under affine transformations. A set function f on a finite set V is submodular if f(X) + f(Y ) ≥ f(X ∪ Y )− f(X ∩ Y ) for any pair X,Y ⊆ V . The symmetric difference transformation (SD-transformation) of f by a canonical set S ⊆ V is a set function g given by g(X) = f(X M S) for X ⊆ V , where X M S...

For a class C of sets, let us say that a set A is C stabilising if A 4 X ∈ C for every X ∈ C. We prove that the Martin-Löf stabilising sets are exactly the K-trivial sets, as are the weakly 2-random stabilising sets. We also show that the 1-generic stabilising sets are exactly the computable sets.

Okhotin [9] showed an exponential trade-off in the conversion from nondeterministic unary finite automata to unambiguous nondeterministic unary finite automata. In this paper, we consider the trade-off in the case of unary symmetric difference finite automata to finitely ambiguous unary symmetric difference finite automata. Surprisingly, the trade-off is linear in the number of states of the fi...