The creation of optimal samplers can be a challenging task, especially in the presence of constraints on the support of parameters. One way of mitigating the severity of this challenge is to work with transformed variables, where the support is more conducive to sampling. In this work, a particular transformation called inversion in a sphere is embedded within the popular Metropolis-Hastings paradigm to effectively sample in such scenarios. The method is illustrated on three domains: the standard simplex (sum-to-one constraint), a sector of an n-sphere, and hypercubes. The method's performance is assessed using simulation studies with comparisons to strategies from existing literature.

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.