By Emil Grosswald, Marvin Isadore Knopp, Mark Sheingorn

Emil Grosswald used to be a mathematician of significant accomplishment and noteworthy breadth of imaginative and prescient. This quantity can pay tribute to the span of his mathematical pursuits, that's mirrored within the wide variety of papers accrued right here. With contributions through major modern researchers in quantity concept, modular features, combinatorics, and similar research, this booklet will curiosity graduate scholars and experts in those fields. The prime quality of the articles and their shut connection to present learn tendencies make this quantity a needs to for any arithmetic library

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All the arcs in G1 not in the circuit of graphs, form a graph I. Let 11 be a connected piece of I. Then 11 has at most a single vertex in common with the rest of G. For suppose 11 had the two vertices c and d in common with H. From c follow along some chain towards d in H till we first reach a vertex e in 11. From e follow back along some chain in 11 to C. We have formed thus a circuit containing arcs of both H and It. But as H consists of a certain subset of the components of G1, this circuit contains arcs of at least two components of G1, contrary to Theorem 17.

We have to prove this new form of our condition by showing that P will completely contain a serial An if and only if it is satisfied by functions of at leaRt one of our six types. Now an alternative in ny's is serial in XO if it contains either (i) or, for short, XO(Yl) . ) . ), r<. or or (d) n . . Po(y,·, r<. ) . , y,). • A result previously obtained for type (5) by Langford, op. cit. 21 F. P. 284 [Dec. 13, RAMSIW rrhere are tllllR altogether eight alternatives serial in uoth "'0 and XO got by combining either of (i), (ii) with any of (a), (b), (c), (d); but these eight serial alternatives only give rise to six serial forms, since the alternatives (i) (b) and (i) (c) can be obtained from one another by reversing the order of the y's and so belong to the same form, and so do the alternatives (ii) (b) and (ii) (c).

Thus the arcs of G1 forming the circuit R are the arcs of the chains C and D. As G1 and G{ contain fewer than E arcs, we can map them together on a * Obviously G, is non-separable. 45 360 HASSLER WHITNEY [April sphere so that properties (1) and (2) hold. a{ lies on one side of the circuit R, which we call the inside. Each arc of R is crossed by an arc on a{ , and thus there are no other arcs of G{ crossing R. There is no part of G{ lying inside R other than a{ , for it could have only this vertex in common with the rest of G{ , and G{ would be separable.