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The graph of a set grammar is introduced in such a way that each set rule of the grammar is represented by a cartesian subgraph of it. The correspondence between cartesian subgraphs and transitions of Petri nets (which satisfy the axiom of extensionality) is established. The set grammars with input (initial) and output (terminal) elements are studied in an analogy to Chomsky's string grammars and their strong equivalence. Permit rules and parallel permit rules are introduced in such a way that parallel permit grammars are more general tools than Petri nets themselves, because the equivalence between homogeneous parallel permit grammars and set grammars (and Petri nets) is proved.

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Let, for each n?N, (X i,n)0 ? i ? nbe a triangular array of stationary, centered, square integrable and associated real valued random variables satisfying the weakly dependence condition lim N ? N 0limsup n ? + 8 nSr=N nCov (X 0,n, X r,n)=0;where N 0is either infinite or the first positive integer Nfor which the limit of the sum nSr=N nCov (X 0,n, X r,n) vanishes as n goes to infinity. The purpose of this paper is to build, from (X i,n)0 ? i ? n, a sequence of independent random variables (X˜i,n)0 ? i ? nsuch that the two sumsSi =1 n X i,nandSi =1 n X˜i,nhave the same asymptotic limiting behavior (in distribution).

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Abstract  

Let G be a finite group. Let p(G) denote the minimal degree of a faithful permutation representation ofG and let q(G) and c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers, respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. The purpose of this paper is to calculate p(G), q(G), c(G) and r(G) for the group SP(4,q).

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