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MoA (A Mathematics of Arrays), is a new theory created in my dissertation, ('88 Syracuse University) and has been used to theorize about optimal mapping to advanced architectures. The power of this theory is that through linear and multi-linear transformations (psi reduction), i.e. index composition, it is possible to define any array algorithm and reduce it to a normal form denoting the least amount of arithmetic and IO needed. Although I have mapped algorithms to parallel and distributed architectures, I have not explored mapping to processors and GPUs. With these resources I would take many algorithms I have derived and studied (Kronecker Product, Khatri-Rao, FFT, Matrix Multiplication, etc.) and theorize about mapping them to GPUS and CPUS. Assuming successful research it will enable my research to move forward to identify more methods of automation where all is verfied. I desire to enable users to specify their algorithms mathematically and through MoA mechanize the entire mapping, incorporating tools such as OPENACC,  OPENMP and other tools where semantics maps to the loops revealed by MoA.

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