the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  wea...

The C-C bond formation is an important reaction in organic synthesis to obtain value-added intermediates. Therefore, in this paper an attempt has been made to accelerate the Henry reaction (C-C bond formation) between aryl aldehydes and nitromethane using less expensive whole cell biocatalyst, baker’s yeast (BY). The scope of the methodology was also tested for the heteryl aldehyde i.e. 2-chlor...

The C-C bond formation is an important reaction in organic synthesis to obtain value-added intermediates. Therefore, in this paper an attempt has been made to accelerate the Henry reaction (C-C bond formation) between aryl aldehydes and nitromethane using less expensive whole cell biocatalyst, baker’s yeast (BY). The scope of the methodology was also tested for the heteryl aldehyde i.e. 2-chlor...

In a computational process, certain entities (for example sets or arrays) and operations on them may be automatically available, for example by being provided by the programming language. We define background classes to formalize this idea, and we study some of their basic properties. The present notion of background class is more general than the one we introduced in an earlier paper, and it t...