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In this paper, for a select group of quadratic algebras, we
investigate restrictions necessary on the generators of the ideal for
the resulting algebra to be Koszul. Techniques include the use of
Grobner bases and development of Koszul resolutions. When the
quadratic algebra is Koszul, we provide the associated linear
resolution of the field. When not Koszul, we describe the maps of
the resolution up to the instance of nonlinearity.