Given a finite dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext 2 C (DC, C) of C by the CC -bimodule Ext 2 C (DC, C). We give a construction for the quiver of the relation-extension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebrã C ...

It reviews the theory of Hidden Filter Hidden Markov Models and presents an extension, Mixed State Hidden Markov Models, developed jointly by Andrew Fraser and myself under his supervision. This manuscript version has only trivial differences from the original.

It is proved that the second cohomology group of the conformal algebras Cendn and Curn with coefficients in any bimodule is trivial. As a result, these algebras are segregated in any extension with a nilpotent kernel.