### Table 1. Trinomials X l + u ? A ; (v) X l + u B ; (v) are reducible. The theorem presents a complete analogy with Capelli apos;s theorem in which there is an exceptional case a = ?4 b4 and an in nite sequence of exceptions corresponding to a = bp. The proof rests on the existence of a lower bound for the genus of a certain algebraic curve except in a nite number of cases. Once this bound is known, the problem is solved by a method of indeterminate coe cients. Schinzel gave a theorem for the algebraic function elds but it is too technical to be cited here and another theorem for the algebraic number elds ( nite extension of Q). The latter is rather complicated and we only comment it. Like the rst Schinzel theorem it gives a criterion to recognize reducible polynomials. The 167

### Table 2 Clustering coeSOcients of the market graph

2004

### Table 1: Event probabilities for causal structures Event Graph 0 Graph 1 Graph 2

2004

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### Table 1: The possibilities for r and n for g = 1; 2; 3; 4; 5

"... In PAGE 14: ... This is the case (2; 2g + 1) corresponding to a one-dimensional solution. Table1 gives an overview of the possibilities for low... In PAGE 16: ... Depending on details of the initial data, step 1 might give the genus of the solution precisely, or it might give a lower bound on the genus. Then for a given genus, step 3 identi es a unique pair (r; n) of stationary ows only if the initial data is one-dimensional (so r = 2); otherwise it gives a nite list of possible pairs of stationary ows, as indicated in Table1 . The rst place to test whether a pair (r; n) of stationary ows is appropriate for given initial data occurs here, at step 5.... In PAGE 20: ... In this case, the KP equation can be integrated directly. On the other hand, Table1 shows that if g = 1 then r = 2 and n = 3. We show next that the BC matrix provides the same information that one obtains from direct integration of the KP equation.... In PAGE 24: ... Example 4. r = 3; n = 4 From Table1 , an (r = 3; n = 4) solution can have either genus 2 or genus 3. A typical (r = 3; n = 4)- potential has genus 3.... In PAGE 30: ...when shortened by these two additional requirements, provides the complete list of pairs (r; n) of stationary ows that can correspond to a KP solution of the form (1), and of genus g. The lists given in Table1 were obtained in just this way. 2 Theorem 2 Pg, the number of possible pairs (r; n) that can correspond to a solution of genus g is bounded by g(g + 1)=2 ? g + 1.... ..."