The Bott–Thurston cocycle is a $$2$$ -cocycle on the group of orientation-preserving diffeomorphisms circle. We introduce and study formal analog cocycle. continuous $$A$$ -automorphisms algebra $$A((t))$$ Laurent series over commutative ring with values in $$A^*$$ invertible elements . prove that central extension given by equivalent to 12-fold Baer sum determinantal when $$\mathbb Q$$ -algebr...

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an א0-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite excha...

cozero maps are generalized forms of cozero elements. two particular cases of cozero maps, slim and regular cozero maps, are significant. in this paper we present methods to construct slim and regular cozero maps from a given  cozero map. the construction of the slim and the regular cozero map from a cozero map are called slimming and regularization of the cozero map, respectively. also, we pro...

Brainstem stroke syndromes are primarily determined by clinical criteria. There are few diagnostic procedures which are of benefit for the evaluation of brainstem ischemic events. Brainstem auditory evoked responses (BAERs) are a new electrophysiologic technique for assessing brainstem function. To evaluate the use of BAERs in patients with brainstem ischemic events, 35 individuals with recent ...