We find necessary and sufficient conditions of irreducibility of vacuum modules over affine Lie algebras and superalgebras. From this we derive conditions of simplicity of minimal W -algebras. Moreover, in the case of the Virasoro and Neveu–Schwarz algebras we obtain explicit formulas for the vacuum determinants. © 2006 Elsevier Inc. All rights reserved.

How are claims about what people ought to do related to claims about what ought to be the case? That is, how are claims about of personal obligation, of the form s ought to φ, related to claims about impersonal obligation, of the form it ought to be the case that p? Many philosophers have held that the former type of claim can be reduced to the latter. In particular, they have held a view known...

in this paper, let $l$ be a completeresiduated lattice, and let {bf set} denote the category of setsand mappings, $lf$-{bf pos} denote the category of $lf$-posets and$lf$-monotone mappings, and $lf$-{bf cslat}$(sqcup)$, $lf$-{bfcslat}$(sqcap)$ denote the category of $lf$-completelattices and $lf$-join-preserving mappings and the category of$lf$-complete lattices and $lf$-meet-preserving mapping...

We derive minimal conditions on a symmetry group of a linearly elastic material that implies its isotropy. A natural setting for the formulation and analysis is provided by the group representation theory where the necessary and sufficient conditions for isotropy are expressed in terms of the irreducibility of certain group representations. We illustrate the abstract results by (re)deriving sev...