Reduced K-theory for Azumaya algebras

Abstract In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps Ki(F)?Ki(D), where Ki are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

Speech Recognition with unknown partial feature corruption - a review of the union model

This paper provides a summary of our studies on robust speech recognition based on a new statistical approach – the probabilistic union model. We consider speech recognition given that part of the acoustic features may be corrupted by noise. The union model is a method for basing the recognition on the clean part of the features, thereby reducing the effect of the noise on recognition. To this end, the union model is similar to the missing feature method. However, the two methods achieve this end through different routes. The missing feature method usually requires the identity of the noisy data for noise removal, while the union model combines the local features based on the union of random events, to reduce the dependence of the model on information about the noise. We previously investigated the applications of the union model to speech recognition involving unknown partial corruption in frequency band, in time duration, and in feature streams. Additionally, a combination of the union model with conventional noise-reduction techniques was studied, as a means of dealing with a mixture of known or trainable noise and unknown unexpected noise. In this paper, a unified review, in the context of dealing with unknown partial feature corruption, is provided into each of these applications, giving the appropriate theory and implementation algorithms, along with an experimental evaluation.