A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.