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A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one parameter we know a regular solution, then what is the nearest value of the parameter for which the solution in the polynomial homotopy is singular? Applying the ratio theorem of Fabry on the solution paths defined by the homotopy, extrapolation methods can accurately locate the nearest singularity. Once the radius of convergence is known, then via a transformation of the continuation parameter, the series expansions of the solution curves will have convergence radius equal to one. To compute all coefficients of the series we propose the quaternion Fourier transform.

This chapter provides an overview of industry codes and standards that apply to SPDs. SPDs are routinely used in facilities that are potentially exposed to voltage or current surges from nearby lightning, utility switching, or other sources; and there are many manufacturers of SPDs and these manufacturers often offer guidance regarding SPD installation ratings and recommended applications. However, industry codes and standards provide limited guidance regarding selection, rating, and application. One limitation with surge protection design is that there is no industry standard that describes what is an acceptable level of surge protection for standard facilities or residential locations. Although industry codes and standards are available that establish standardized surge criteria and assist with the application of specific surge protector types, these standards do not provide adequate design guidance that ensures a facility is properly protected against surges. There is no existing industry guidance for surge protection of residential facilities.

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