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In this talk we shall discuss the notion of nonnegative polynomial functions.Unlike their quadratic counterpart,nonnegative polynomial functions are not uniquely defined,nor are they easily computable.In this talk we will discuss 6 different cones of nonnegative quartic polynomial functions generated from nonnegative quartic polynomial functions.These convex cones are indecreasing order,much like the Russian Matryoshka dolls,with varying computational complexities.We discuss the modeling power and applications of these convex cones.