Sentence examples for implies the first from inspiring English sources

This paper's result therefore implies the first randomized O(NlogM logθ+1)) solution for the threshold version of the problem.

This result implies the first explicit lower bound on the approximability of the general version (i.e., for arbitrary graphs) of the problem.

This implies the first genuine fragmentation effect that transcends the phylogeny of plants and is not undermined by statistical problems of pseudoreplication.

Since the second relation implies the first one, we also consider the universal C⁎-algebra T⁎ϑT generated by two isometries u and v with the weaker relation uv= e2πiϑvu.

which implies the first inequality.

which implies the first inequality of the claim.

(3) "If one proposition (p) implies a second (q), then if that second proposition implies a third (r), the first implies the third".

which implies the first-order difference inequality (2.9).

Moreover, ∑ i = 1 N U i ( x i, k ) has a continuous first partial derivation, which implies the first-order Kuhn-Tucker conditions are necessary and sufficient for optimality[13].

This also implies the second inequality in the statement.

It remains to show how the first part of the theorem implies the second one.