We give details of the 1-1 correspondence between equiangular frames of nn vectors for RdRd and graphs with nn vertices. This has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory. The nontight examples given here (which correspond to graphs with more than 2 eigenvalues) have the potential for similar applications, e.g., the frame corresponding to the 5-cycle graph Uggs Boots Outlet Uk
is the unique Grassmannian frame of 5 vectors in openR3openR3. Further, the associated canonical tight frames have a small number of angles in many cases.
In this article, Cauchy’s integral formula Genuine Uk Ugg Boots
for nth q-derivative of analytic functions is established and used to introduce a new proof to q-Taylor series by means of using the residue calculus in the complex analysis. Some theorems related to this formula are presented. A q-extension of a Laurent expansion is derived and proved by means of using Cauchy’s integral formula for a function, which is analytic on a ring-shaped region bounded by two concentric circles. Three illustrative examples are presented to be as applications for a q-Laurent expansion.