On some algebraic properties of automata

Let A be a class of Moore automata. It is shown that R(H(S(A))) is closed for the three operators S, H, R where S, H, R denote that the set of subautomata, of factor automata, of the automata obtained by input reduction (respectively) are formed

A criterion for the simplicity of finite Moore automata

A Moore automaton A = (A, X,Y,S, A) can be obtained in two steps: first we consider the triplet (A, X, 6) - called a semiautomaton and denoted by S — and then we add the components Y and A which concern the output functioning. Our approach is: S is supposed to be fixed, we vary A in any possible way, and - among the resulting automata - we want to separate the simple and the nonsimple ones from each other. This task is treated by combinatorial methods. Concerning the efficiency of the procedure, we note that it uses a semiautomaton having |A|(|A| + l)/2 states

Euler characteristics of Hilbert schemes of points on simple surface singularities

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.Comment: 57 pages, final version. To appear in European Journal of Mathematic