On Projective Modules and Computation of Dimension of a Module over Laurent Polynomial Ring

Abstract

We give a procedure and describe an algorithm to compute the dimension of a module over Laurent polynomial ring. We prove the cancellation theorems for projective modules and also prove the qualitative version of Laurent polynomial analogue of Horrocks' Theorem.

Plasmonic Fluorescent Nanocomposites of Cyanines Self-assembled upon Gold Nanoparticle Scaffolds

“…It is known that even within the family of cyanine dyes, certain cyanines form fluorescent J-aggregates whereas others do not. 3,10 Therefore, it is not surprising that eosin, the dye used by Ghosh et al, 25 which is chemically different from our cyanines, might exhibit AuNPs size-dependent formation of fluorescent J-aggregates. Ghosh et al 25 also used a single concentration of AuNP and eosin for their determinations, whereas we concluded size preferences based on titrations involving several different concentrations of the cyanine (vide infra) and the AuNPs (Fig.…”

“…2 Cyanine dyes are known to form fluorescent J-aggregates upon a wide variety of scaffold materials. 3 During a literature review of chromophore assemblies on metal nanostructures, we found several papers reporting no fluorescent J-aggregates being formed or the fluorescence from such aggregates was quenched or photobleached. [4][5][6][7][8][9][10][11][12][13] Others reported a need for spacer layers in order to distance the aggregates from the noble nanometal surface for avoiding quench; 10,12,14,15 sometimes, even with a spacer layer no J-aggregation was observed with certain cyanines.…”