We study the action on modulation spaces of Fourier multipliers with symbols <!-- MATH $e^{i\mu(\xi)}$ --> , for real-valued functions having unbounded second derivatives. In a simplified form our result reads as follows: if satisfies the usual symbol estimates of order <!-- MATH $\alpha\geq2$ --> , or if is a positively homogeneous function of degree , then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces and , for all <!-- MATH $1\leq p,q\leq\infty$ --> and <!-- MATH $s\geq (\alpha-2)n|{1/p}-1/2|$ --> . Here represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function whose Hessian matrix is non-degenerate at some point.

A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presented. The analysis, that is based on the Sobolev space mapping properties of the EFIE, explains the conditioning growth of the majority of currently available quasi-Helmholtz decomposition methods (such as Loop-Tree/Star, Loop-Rearranged Trees, etc.) that cure the EFIE low-frequency breakdown. It is shown that these methods have a conditioning that grows polynomially with the number of unknowns. To solve this problem, in this work we present a quasi-Helmholtz decomposition method that leads to an EFIE whose conditioning grows only logaritmically with the number of unknowns. This result is obtained by properly regularizing both the solenoidal and non-solenoidal part of the EFIE. The regularization is obtained by introducing a new set of loop hierarchical basis functions. Numerical tests are provided to confirm the results obtained by the theory.

Many initiatives exist to increase the adoption of open educational practices (OEP) within universities, but few initiatives start by exploring the capacity of educators to adopt open approaches. This paper addresses this challenge, suggesting that in order to build OEP capacity, universities should build on the existing skills of local champions who are familiar with open approaches. The paper builds on the Open Educators’ Factory methodology to map the capacities of university teachers across four areas: open design, open content, open teaching and open assessment and presents the results of its application to a case study within an Italian university. The pilot demonstrates that by using this approach, it is possible to map universities’ existing OEP and connect them with the capability of local educators. This enables university managers to build on the expertise of open education practitioners to raise the overall capacity of their staff to adopt open approaches.