The Grassmannian and the Lax pair approaches to the Kadomtsev-Petviashvili (KP) hierarchy are analyzed in the framework of (formal) principal fiber bundles. The underlying factorization problem is formulated as a local triviality condition. In particular the common object in the two approaches — the Baker function — coordinatizes the base space, the fiber consists of certain generalized differential operators.
An example is constructed to show that despite the apparent similarity to the splitting in [8], the local triviality condition cannot be given a group theoretic interpretation.

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Let us consider an application of forward local C^{1, ε}-semimartingale flows of C¹-diffeomorphisms. First a change of variable formula is derived and the existence of all moments of the appearing Jacobian is shown. Then as a consequence, Ito's formula holds for continuous functions which first and second order derivatives exist only in the sense of distributions.

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We bring some descriptive-set-theoretical problems into complexity theory. We here deal with the uniformization problem and the separation problem. It is shown that 1) there exists an oracle A such that for some set S∈P[A] the uniformizator U_s is not in NP[A], 2) there is an oracle A such that Sep (NP[A]) does not hold and hence so does not Unif (coNP[A]), and 3) there is an oracle A such that Sep(NEXT [A]) does not hold and hence so does not Unif(coNEXT [A]).

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