Fixed Polarity Pascal Transforms with Symbolic Computer Algebra Applications The fixed polarity forms of the ReedMuller (RM) transform exist in 2n different polarities. The integervalued Pascal transform is related to the binaryvalued RM transform through the Sierpinski fractal, calculated by performing the modulo2 operation on Pascalxe2x80x99s triangle, as it appears in the lower triangular portion of the positivepolarity RM transform. We generalize the relationship between the fixedpolarity forms of the RM transform and introduce associated forms of the Pascal transform that are characterized by a polarity value allowing for a family of fixedpolarity Pascal (FPP) transform matrices to be defined. We observe and prove several properties of the FPP transforms and their inverses. An application of FPP transforms in the area of symbolic computer algebra that enables very fast decomposition of realvalued polynomials as weighted sums of different binomials raised to a power as compared to manual symbolic manipulation is described. The decomposition weights can be considered to be the inverse FPP spectrum with respect to a realvalued polynomial since they are computed using one of the linear orthogonal FPP transformation matrices.

A Hybrid Method for Spectral Translation Equivalent Boolean Functions The equivalence of Boolean functions with respect to five invariance (aka translation) operations has been well considered with respect to the RademacherWalsh spectral domain. In this paper, we introduce a hybrid approach that uses both the ReedMuller and the RademacherWalsh spectra. A novel hybrid algorithm that maps a Boolean function to a representative function for the equivalence class containing the original function is presented. The algorithm can be used to determine a sequence of translations that maps one function to an equivalent function. We present experimental results that show the hybrid algorithm can determine the equivalence classes for 5 variables much more efficiently than before. We also show that for 6 variables where there are 150,357 equivalence classes, 8 are very difficult, a further 58 are difficult and the remainder are straightforward in terms of the CPU time required by the hybrid algorithm.