Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H}$ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all vertices in a particular subset of $ V(\mathcal{H})$, by using an additional set of edges; in particular, we ensure that our matching avoids all additional conflicts, which may consist of both old and new edges. This setup is useful for various applications in design theory and Ramsey theory. For example, our main result provides a crucial tool in the recent proof of the high-girth existence conjecture due to Delcourt and Postle. It also provides a black box which encapsulates many long and tedious calculations, greatly simplifying the proofs of results in generalised Ramsey theory.

Continuous immersion in a second language causes speakers’ first language to change, a phenomenon known as L1 attrition. We explored (1) whether bilingual native Mandarin speakers display attrition-related changes in their use of referring expressions in Mandarin after exposure to English and (2) whether the severity of attrition is affected by the amount of exposure to both Mandarin (L1) and English (L2) and English proficiency. All participants completed a questionnaire to assess their language experience and a picture description task in spoken Mandarin. The results show that where more monolingual Mandarin speakers preferred null pronouns, bilingual speakers tended to use overt pronouns, suggesting attrition-related changes in their native language which favoured explicitness. Our study also shows that decreased use of L1 coupled with increased use of L2 and higher L2 proficiency are likely to result in a greater degree of attrition, although such an association is statistically unreliable in some models.

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ($f$) and decreases with the aspect ratio ($\epsilon$), following a $\epsilon f^{-1}$ power law. Using a three-dimensional (3-D) initial value and two-dimensional eigenvalue calculations, we identify a dominant 3-D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode’s frequency and transverse wavenumber as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.