Cruising The Simplex: Hamiltonian Monte Carlo and the Dirichlet Distribution

Due to its constrained support, the Dirichlet distribution is uniquely suited to many applications. The constraints that make it powerful, however, can also hinder practical implementations, particularly those utilizing Markov Chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo. I demonstrate a series of transformations that reshape the canonical Dirichlet distribution into a form much more amenable to MCMC algorithms.Comment: 5 pages, 0 figure

Local cohomology properties of direct summands

In this article, we prove that if $R\to S$ is a homomorphism of Noetherian rings that splits, then for every $i\geq 0$ and ideal $I\subset R$, \Ass_R H^i_I(R) is finite when \Ass_S H^i_{IS}(S) is finite. In addition, if $S$ is a Cohen-Macaulay ring that is finitely generated as an $R$-module, such that all the Bass numbers of $H^i_{IS}(S)$, as an $S$-module, are finite, then all the Bass numbers of $H^i_{I}(R)$, as an $R$-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik. As a consequence, we exhibit a Gorenstein $F$-regular UFD of positive characteristic that is not a direct summand, not even a pure subring, of any regular ring.Comment: 8 pages. References updated. Minor change