By Jost J., Xin Y. L.
We receive a Bernstein theorem for exact Lagrangian graphs in for arbitrary basically assuming bounded slope yet no quantitative restrict.
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Extra resources for A Bernstein theorem for special Lagrangian graphs
39) n=0 We next obtain a relation between GA (z) and FA (z) that will enable us to determine both. If b1 b2 · · · bn is counted by fA (n), then for x either 0 or 1, the string xb 1 b2 · · · bn either does not contain A at all, or if it does contain it, then A = xb 1 b2 · · · bk−1 . 40) and multiplying both sides of Eq. 40) by z n and summing on n ≥ 0 yields 2FA (z) = z −1 (FA (z) − 1) + z −1 GA (z) . 41) We need one more relation, and to obtain it we consider any string B = b 1 b2 · · · bn that does not contain A any place inside.
For example, the exponential generating function for derangements (Eq. 6) is ∞ f (z) = dn n z = n! n=0 ∞ = k=0 ∞ (−1)k k! n=k ∞ zn n! n=0 zn = n (−1)k n=0 e−z 1−z , n! k! 7) which is extremely compact. Reasons for using generating functions go far beyond simplicity. The one that matters most for this chapter in that generating functions can be used to obtain information about the asymptotic behavior of sequences they encode, information that often cannot be obtained in any other way, or not as easily.
5) for 1 ≤ j ≤ 3. This holds because the fj (z) have the same behavior as z → 1− , and is part of a more general phenomenon. Good knowledge of the behavior of the generating function on the real axis combined with weak restrictions on the coefficients often leads to estimates for the summatory function of the coefficients. There are cases where elementary methods give precise bounds for individual coefficients. 6) ∞ g(z) = n=0 Then for gn z n = (1 − z)f (z) . 8) k=0 and so estimates of the summatory function of the g k yield estimates for fn .