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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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The aim of this paper is to find a closed form of the integrals ∫ 0 π cos( x sin( t ) − nt ) d t for n = 0, 1, 2, … using the Maple computer algebra system. Although Maple 10 is not capable to calculate these integrals in one step, it turns out to be a very useful tool to solve this and similar kind of complex mathematical problems. During the problem solving process Maple proves that it is useful and, what is more, it is an indispensable partner. Maple helps us to formulate our conjecture, acts as an advisor and, last but not least, performs complex symbolic calculation instead of us.

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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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