Picard's little theorem

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Picard's little theorem

[pi′kärz ′lid·əl ¦thir·əm]
(mathematics)
A nonconstant entire function of the complex plane assumes every value save at most one. Also known as Picard's first theorem.
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Therefore, the Picard theorem implies that, for each 0 < [epsilon] < 1, there exists a unique solution [S.sub.[epsilon]](t) [member of] [C.sup.1] (0, [T.sub.[epsilon]];[X.sup.m]) for a fixed time [T.sub.[epsilon]] > 0.
To use the Picard theorem on [H.sup.m] space (m > 5/2), we first obtain that F is Lipschitz continuous on [H.sup.m], i.e.,
Ullrich teaches readers how to think like analysts long before they get to the "Big Picard Theorem," explaining why proofs do and do not work.
Green, On functional equation [f.sup.2] = [e.sup.2[[Phi].sub.1]] + [e.sup.2[[Phi].sub.2]] + [e.sup.2[[Phi].sub.3]] and a new Picard theorem, Trans.
[7] , Some Picard theorems for holomorphic maps to algebraic varieties, Amer.
The "Hyperbolically imbedded in Y" is a very important property because it implies many properties, such as the finiteness of integral points (as we have mentioned in the Lang's conjecture), extension properties (big Picard theorems), etc.