We present an overview of the notions exact sequences Hopf algebras and tensor categories their connections. also some examples illustrating main features; these include simple fusion a natural question regarding composition series finite categories.

We consider exact algorithms for Bayesian inference with model selection priors (including spike-and-slab priors) in the sparse normal sequence model. Because best existing algorithm becomes numerically unstable sample sizes over n=500, there has been much attention alternative approaches like approximate (Gibbs sampling, variational Bayes, etc.), shrinkage (e.g. Horseshoe prior and Spike-and-S...

Let Λ be a quasi k-Gorenstein ring. For each dth syzygy module M in mod Λ (where 0 ≤ d ≤ k − 1), we obtain an exact sequence 0 → B → M ⊕ P → C → 0 in mod Λ with the properties that it is dual exact, P is projective, C is a (d + 1)st syzygy module, B is a dth syzygy of Ext Λ (D(M),Λ) and the right projective dimension of B ∗ is less than or equal to d − 1. We then give some applications of such ...