We review Fujiwara's scheme, a sixth order weak approximation scheme for the numerical approximation of SDEs, and embed it into a general method to construct weak approximation schemes of order 2m for m ∈ N. Those schemes cannot be seen as cubature schemes, but rather as universal ways how to extrapolate from a lower order weak approximation scheme, namely the Ninomiya-Victoir scheme, for higher orders.
The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded test functions. We first describe a recently introduced flexible functional analytic framework, so called weighted spaces, where Feller-like properties hold. A refined analysis of vector fields on weighted spaces then yields optimal convergence rates of cubature methods for stochastic partial differential equations of Da PratoZabczyk type. The ubiquitous stability for the local approximation operator within the functional analytic setting is proved for SPDEs, however, in the infinite dimensional case we need a newly introduced technical assumption on weak symmetry of the cubature formula. Computational results for a cubature discretization of a spatially extended stochastic FitzHugh-Nagumo model, an SPDE model from mathematical biology, are shown, illustrating the applicability of our theory.
In [Comm. Algebra 29 (2001) 193-201] Cimprič gave examples of division rings containing an ordering of level 2m but not of level m for m ∈ N. His examples were quite complicated. We give substantially simplified examples in Section 2. In Sections 3 and 4 we investigate this phenomenon using valuation theory. We define almost real and n-real valuations and study liftings of orderings from the residue division ring to the original division ring. Such liftings are not always possible (as is the case in the commutative setting), but we give a necessary and sufficient condition for a lifting to exist. We also prove a suitable generalization of the Baer-Krull theorem. Finally, in the last section we use our examples and the theory developed to answer a question given by Marshall and Zhang [
The n-th product level of a skew-field D, ps n (D), is a generalization of the n-th level of a field F , s n (F ). An explicit bound for s 2m (F ) in terms of m and s 2 (F ) is known and it is also known that there is no such bound for ps 2m (D) when m is even. Our aim is to explicitly construct such a bound for odd m. More precisely, we construct a function f : N 3 → N, such that ps 2 k l (D) f (ps 2 k (D), k, l), for every integer k, every odd integer l and every skew-field D.We give an explicit bound for the n-th Pythagorean number of a simple extension Q(d)/Q in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n-th product level of an Ore domain is equal to the n-th product level of its skew field of fractions.
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