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It is known how to obtain a uniform estimate of e.g. a polynomial in terms of its logarithmic sum over the integers provided that the sum is sufficiently small. This result is generalized here and we obtain estimates in terms of logarithmic sums taken over certain discrete subsets of the real axis.

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Characteristic numbers of families of maps of nodal curves to$P$^{2}are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.

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